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PHYSICAL LABORATORY NOTES. 



MASSACHUSETTS INSTITUTE OF TECHNOLOGY. 



7 



BY 



Y 



SILAS Wl'HOLMAN, 

Associate Professor of Physics. 




BOSTON : 

J. S. GUSHING & CO., PRINTERS. 

1885. 



PREFACE. 



r I ^HESE notes are a portion of the directions which have been prepared from 
-*- time to time during the past ten years as an aid in class instruction at the 
Rogers Laboratory of Physics of the Massachusetts Institute of Technology. 
They are now printed (but not published) by the Institute for the convenience 
of its students. About one-half of the notes up to page 3G are supplementary 
to directions given in Pickering's "Physical Manipulation," vol. i., but the 
greater portion of the remainder is entirely independent of any text-book. 

The treatment of the subjects of Thermometry and Heat-Measurements is 
original. The experiments there described have been found, by experience 
with classes, to be satisfactory both as to accuracy of results and value for 
training in careful manipulation. Few, if any, existing text-books give an 
exact statement of the proper use of the mercurial thermometer with an. 
accuracy of 0.1° to 0.01° C. below 100° C. In view of this fact, and of the 
necessity of this knowledge in many branches of professional work, it is 
thought that the section on Thermometry will be of special interest. The 
omissions in regard to the use of very sensitive thermometers and to measure- 
ments of very high temperature are intentional, these points being considered 
beyond the scope of this class of laboratory instruction. The notes on Elec- 
trical Measurements are designed rather for students who are beginning a 
somewhat elaborate course in that subject than for general students. The 
notes on Photography are by Mr. W. H. Pickering, Instructor in Physics 
at the Institute. 

It will be seen that, throughout the course, the work is quantitative. In 
connection with the later portions, and with the more advanced work which 
follows, attention is directed to the study of all sources of error and to the 
proper adjustment of the relative precision in the component parts of complex 

measurements. ♦ 

SILAS W. HOLMAN. 

Do or. Oc >ber, 1885. 






CONTENTS. 



MECHANICS. 

PAGE. 

Verniers J 

Insertion of Cross-Hairs. (Pickering, i. 29.) 2 

Eccentricity of Circles 2 

Spirit Level 4 

Estimation of Time of Transit . • 6 

Method of Weighing 7 

Adjusting- and Testing a Balance 9 

Meteorological Instruments 13 

Composition of Forces .16 

Dividing Engine. (Pickering, i. 56.) 17 

Crank Motion. (Pickering, i. 68.) 17 

Angle of Friction. (Pickering, i. 71.) . 18 

Breaking Weight 18 

Deflection of Beams, I. (Pickering, i. 77.) 19 

Deflection of Beams, II 20 

Law of Pendulum 21 

Metronome Pendulum . . . . • 21 

Borda's Pendulum (Pickering, i. 85.) 23 

Torsion Pendulum 24 

Volume of Flask 21 

Specific Gravity of Solids 25 

Specific Gravity of Liquids 28 

Mariotte's Law (a) 29 

Mario tte's Law (&)... 30 

LIGHT. 

Bunsen Photometer 31 

Absorption Spectra 34 



CONTENTS. 

PAGE. 

Index of Refraction. (Pickering, i. 141.) 35 

Law of Refraction 80 

Photography 85 



HEAT. 

Thermometry . . y . . . ' .37 

r 

Specific Heat .............. 44 

Latent Heat of Vaporization 48 

Expansion of Liquids 51 

Expansion of Gases 53 

Adjustments of the Cathetometer 57 

ELECTRICITY. 

Magnetic Inclination or Dip 61 

Use of Galvanoscopes 62 

Measurement of Resistance by Substitution 04 

Differential Galvanometer 00 

Slide-Wire Bridge . 07 

Wheatstone's Bridge . .08 

Temperature Corrections in Resistance Measurements 73 

Specific Resistance 74 

Coefficient of Torsion 75 

Law and Factor of Galvanometer . 70 

E. M. F. and R. of Battery 82 



PHYSICAL LABORATORY NOTES. 



VERNIERS. 



The object of a vernier is to read correctly and with facility the position 
which a reference mark occupies in regard to a graduated scale, e.g. the posi- 
tion of the mark 0, Fig. 1, beside the scale AB. By eye estimation, this might 
be read as 11.6 plus a certain fraction of ac, which might be estimated 
as about 0.3 of ac; so that the reading would be 11.63. The vernier 
is designed to give this fraction with greater certainty, and with- 
out the estimation by the eye. For this purpose there is laid off upon 
the sliding part CF, Fig. 2, carrying the reference mark, a length de 
equal to a certain number m of the smallest scale divisions ; and 
this length de is divided into one more (or one less) parts than 
m. In the second figure, CF represents the first vernier of the 
series used in the experiment, and m = 9, de = 10. Thus each 131 

in 9 Fig 1 

division of the vernier is — = — of a scale division. Thus, if the 

de 10 
line 0, i.e. the zero-point of the vernier, be made to coincide with any line, 
e.g. 8, of the scale, the next vernier division, 1, will fall below G, the next 
scale division, by T ^ of the distance between 8 and G. Conse- 



IT* 

~~ s 

.H — Z 

G — — 
g_ __U 

v -5P 



quently if the vernier be moved until the line 1 coincides with the 
line G, the zero-point will have been advanced 0.1 of 678 beyond 8, 
but G8 = 0.1 of the scale unit (i.e. of the space between 8 and 9 on 
the figure); hence the vernier has been moved 0.01 of a scale unit past 
D, and the space (corresponding to ao in Fig. 1; which we desire to 
know is 0.01 of a unit, the whole reading thus being, in the supposed 
case, 8.01. If the line 2 were made to coincide with F, the reading 
would be 8.02, and so on. In the same way with any other 
vernier, find how many divisions m upon the scale are com- 
prised in the whole length of the vernier, and into how many 

parts n the vernier is divided. Then the vernier division is one — th of a scale- 

n 

division, and the difference in length between a vernier and a scale division is 
-th of a scale division. Find, next, into how many parts p a scale unit is- 
divided. Then the vernier division will be shorter (or longer) than a scale 

Hence, if the first division of the vernier 



Fig. 2;.. 



coincides with a scale division, the space ao is 



cides, ao — 2 



In the first vernier of the series, m 



if the second line coin- 



division by - X - of a scale unit. 
J n p 

1 x 1 ' 

n p J 
- X - ), and so on, as shown numerically in the example just given. 

9, n=10, andi> = 10; thus - X 1=0.01. 
n p 
and the vernier reads to 0.01 of a scale unit, as already shown. In the third 
vernier, m = 24, n = 25, p = 20, and the vernier thus reads to fa X fa — jio = 0.002. 
To read a vernier, having already determined the fraction to which it reads, take 
first the whole of the scale reading up to the line a, Fig. 1, next preceding the 



2 INSERTION OF CROSS HAIRS. 

zero-point of the vernier ; then follow along up the vernier to the line which is 
just opposite one on the scale. Multiply the number of this line from the zero 
by the fraction to which the vernier reads, and add this to the scale reading. 
The vernier is, however, usually so numbered as to render this multiplication 
unnecessary, and the following is a somewhat simpler method of reading, which 
may be adopted when the principle of the instrument is understood. Consider 
the vernier as simply a magnified and subdivided scale division.' Thus, in 
the first vernier, the whole length of the vernier would be a representation of 
0.1 of a scale unit, i.e. of one of the smallest scale divisions magnified and 
subdivided into ten parts ; so that the vernier reads to 0.01 of a unit. In 
reading, take first the whole scale reading to the line next preceding the zero of 
the vernier; then append to this decimal place the next as read from the vernier 
(or acid the vernier reading, if other than a decimal), by noting the line coincid- 
ing with one on the scale. The numbering on the vernier should show at a 
glance the fraction to which it reads. When no line of the vernier is exactly in 
coincidence with one on the scale, the line nearest to coincidence is read ; or the 
precision may be increased by estimation. 

Become familiar, by repeated setting, with each of the verniers in the block 
used. Then set the verniers to the following readings: 1st, 8.03; 2d, 29.9; 3d, 
30.866; 4th, 4° 10' ; 5th, 0° IV ; 6th, 2° 58' 30"; 7th, 48° 52'. Then allow an in- 
structor to see that the settings are correct, and to set the verniers to be read. 



INSERTION OF CROSS HAIRS. 

Read "Physical Manipulation," i. pp. 29-31. 

Use thin card-board. Draw two fine lines, intersecting at C, at right angles 
to each other. With C as a centre, describe an outer circle of a diameter slightly 
smaller than the inside diameter of the tube, and an inner circle 
a little larger than the aperture of the diaphragm within the 
tube. Cut out this ring, and, by placing it 'in the tube in the 
position it will occupy when in use, see that it fits without bind- 
ing. Pin the ring down to the papered side of the block at points 
A and B. At the point Z), on the extension of one of the lines 
intersecting at C, insert a pin. Draw out two silk fibres, and to 
the ends of each attach small bits of paper, so that they may 
be easily seen and picked up. Wind one end of the fibre about 
the base of the pin at D, and stretch it across the ring over one 
Fig. 3. f the intersecting lines. Holding it in this position, secure the 
fibre at L by a piece of mucilaged paper. Then with strips of 
mucilagecl paper secure the fibre in its place upon the ring at F and H. Cut 
the fibre close to the ring, and, moving pin B to K, proceed in the same way to 
attach the cross fibre. The projecting ends of the paper strips should be 
allowed to remain. They serve, by their friction upon the inner surface of the 
tube, to hold the ring in place. 



ECCENTRICITY OF CIRCLES. 

In the use of any such instrument as the theodolite, compass, goniometer, 
etc., where angles are to be read from a graduated circle by means of verniers 
or other devices, and when the full accuracy of which the circle is capable is 
desired, correction must always be made for the eccentricity of the mounting, 
unless this be proved so small as to introduce only a negligible error. The 





ECCENTRICITY OF CIRCLES. 3 

eccentricity arises from the difficulty of so constructing the instrument that the 
axis of rotation of the plate to which the verniers are attached shall pass exactly 
through the centre of the graduated circle. Thus in the figure let A and B 
represent the zero-points of a pair of verniers intended to be 
at opposite ends of a diameter of the circle; but suppose AB, 
by imperfect construction, to pass on one side of the centre 
of graduation C. Suppose also that C is the axis about 
which rotates the plate to which A and B are rigidly at- 
tached. This line AB may or may not pass through C; in 
general it does not. If C coincide with C, there will be no 
eccentricity, and one vernier will suffice. The use of the Fig. 4. 

second vernier is simply to eliminate the error due to almost 
unavoidable eccentricity. The nature of this error can be seen by an inspection 
of the figure, and will be seen to depend on the distance of C from C, ancl of the 
line AB from C, the former causing a periodic, the latter a constant error. 

Suppose the verniers are in any such position as at A and B in the figure, and 
call A ancl B their respective observed readings. The line A'B' is drawn 
through the centre of graduation C, ancl parallel to AB. Then A' and B' are the 
true readings corresponding to this setting of the instrument, for these are the 
readings which A and B would have if AB passed through ancl revolved about 
C, the proper centre. Now it is necessary to find from A ancl B the true reading 
of one or the other vernier, say the reading A'. From B draw through C the 
straight line BCD. Remembering that with the numbering of the circle as shown 
in the figure, the reading B is the length of the arc 0, A, 90, 180, B, it will be seen 
that B - 180° = 02). Also from the figure, A'D = A' A. Therefore B - 180° is 
as much smaller than the true reading A' as the observed reading A is greater 
than A' ; or vice versa if AB happens to be on the other side of A'B'. Thus in 
any case, until B passes the 0° point, that is, so long as B is numerically greater 
than A, the corrected reading of vernier A will be the mean of B — 180° and A, 
ancl may therefore be found either by the expression A' = %(A + B — 180°), or, 
by the rule, subtract 180° from the reading of B, and find the mean between this 
difference and A. 

When B is numerically less than A, that is, after B has revolved past the 
zero-point, it becomes necessary to acid 180° to B instead of subtracting, as will 
be readily seen from a figure drawn to suit this case. Insert the figure and 
demonstration for this case in the note-book. 

With the circle used in this experiment, start with A at about 0°, and 
read both A and B. Turn A to about 20°, and read both verniers as before. 
Continue at intervals of about 20° throughout the whole circumference. After- 
ward compute the true reading for A in each of these positions. 

To study the nature of this error, compute the error of A at each observed 
reading, and construct a curve with readings of A as abscissas, and errors as ordi- 
nates. If AB pass through C, this curve will be half above and half below the 
axis. In general this is not the case. State in the note-book what would be the 
form of the curve, and its position in regard to the axis : first, when AB passes 
through C, but C and C do not coincide ; second, when AB does not pass 
through C, but C and C are coincident. 

Circles are frequently numbered from one zero-point both ways to 180°. 
An inspection of the figure will show that for this case 180° — B will equal 0D. 
Hence J [A + (180° — Z?)] gives A'. Ancl from this it is also evident that when, 
as in compass circles, the numbering is up to 90° in each of the four quadrants, 
running both ways from two zero-points which are at opposite ends of a diame- 
ter, it is only necessary to take the mean of A ancl B ; that is, A 1 = \ (A + B). 



m 



SPIEIT LEVEL. 



SPIRIT LEVEL. 

The spirit-level consists of a glass tube AB, of which the upper surface is 

so curved that a longitudinal section would be 
the arc of a circle of several feet radius, the 
sensitiveness of the level being greater as the 
*23 radius of curvature is greater. The tube is 



3£ 



[j 



FlG> 5> nearly filled with pure alcohol, or a mixture of 

alcohol and a small proportion of sulphuric ether, 
the latter being a more mobile liquid, rendering the bubble more prompt and sure 
in action. On account of its greater expansibility, ether cannot be used alone 
unless the tube is provided with a suitable expansion chamber at the end con- 
taining a bubble of considerable size. 

The tendency of a free liquid surface to become a plane perpendicular to the 
line of the earth's attraction causes the bubble always to approach that point of 
the arc at which such a plane would be tangent to it. The direction of the 
plumb line would be always perpendicular to this plane, which would of course 
deviate from the horizontal just as much, and by the same causes, as the plumb 
line from the vertical, — often a measurable amount. 

The tube when mounted is usually provided at one end with an adjusting 
screw, by which the lower surface or base of the level may be rendered parallel 1 
to the tangent plane through the centre division o of the tube ; that is, so that 
when CD is placed on a horizontal surface, the ends of the bubble may be made 
equidistant from the centre line. 

To make the adjustment, place the level on any nearly horizontal and quite 
smooth surface, as a plate of ground glass, and read the position of each end of 
the bubble, calling these a and b respectively. Note carefully, or mark on the 
plate, the positions of the ends of the level, and then reverse it as exactly as 
possible. Call a' and b' the readings in the new position. In what follows it is 
assumed that the numbering upon the level is from the centre o in each direction, 
and that the readings a, a', etc., and b, b', etc., are respectively in the directions 
oA and oB, the ends A and B being permanently lettered on the instrument. 

A consideration of the conditions will show that if b' — a and a' — b the level 
is in proper adjustment, otherwise the screw at one end must be turned in the 
proper direction, without moving the whole instrument, until the bubble is made 
to take a position at either end about half way between the positions b' and a 
or a' and b. This operation must be repeated until the same readings (&' = a and 
a' — b) are obtained upon reversal. If, however, the plate is not approximately 
level (the degree of approximation depending on the sensitiveness of the level), 
it must first be made so by the use of the method described in the following 
paragraph. Also, if the level is so far out of adjustment as to cause one end of 
the bubble to disappear beneath the metal at one end of the tube, the rule in 
regard to halving the differences of positions between b' and a, etc., is not to be 
followed, since the curvature of the glass towards the end of the level is greater 
than in the middle, and consequently, the adjustment would be accomplished by 
a motion of the adjusting screw which would move the visible end of the bubble 
much less than half the distance between b' and a. In any such case two or 
three approximate adjustments will be requisite. 

Levels attached to instruments are, for convenience, often adjusted and 
permanently fixed in position, but must usually be adjusted at each time of use. 
Adjustment is not, however, always essential, for if an unadjusted level be 
placed upon a horizontal plane surface, the reading of the end of the bubble will 



SPIRIT LEVEL. 5 

remain unchanged, whatever the position of the level upon the surface. Hence, 
to level with an unadjusted level, it is only necessary to move the surface (by 
means of the levelling screws provided) until, on being turned into auy position 
on the surface, the level shows no change of reading. Of course a plane surface 
will always be level when two lines at right angles upon it are level ; and where, 
as is usual, a surface has three levelling screws, the quickest manner of 
levelling it is to level one line parallel to the line joining two of the screws, and 
another line at right angles to this. Adjust the level, level the glass plate, and 
test the levelling of the plate by means of the level when thrown slightly out of 
adjustment. Eecord all the numerical readings taken in the adjustment of the 
level as described, and also by figures demonstrate the method of adjusting the 
level by reversion on an approximately level surface, and the method of levelling 
with an instrument out of adjustment. 

On engineering and astronomical instruments it is sometimes necessary, 
either for the determination of errors or for the accurate measurement of small 
vertical angles, to know with precision the angular displacement of the level 
corresponding to the motion of the bubble over one division of the scale on the 
tube. This quantity may be measured as follows, and in sensitive levels may be 
as small as a few seconds of arc. 

Upon a firm table FG is placed a rigid bar AB supported at the end A by 
two projecting points which rest on a plate _ j r __ nJ9 



of wood or metal, and carrying at the end B 

a micrometer screw whose point rests on a | ^ c cf \ 

flat plate (7, of metal or glass! The distance Fm - 6 - 

from the point of the micrometer to the line joining the points at A should be 
measured. The degree of precision of this measurement should depend on the 
precision with which the bubble can be twice set at the same point in the tube ; 
and as this will seldom exceed or equal the hundredth part of a division, the 
measurements of AC, and by the micrometer, should be made to about 0.1 per 
cent. The level is placed on the bar, preferably near one end. The micrometer 
is turned until the end of the bubble is at any desired mark (say 5), and the 
readings of both micrometer and level recorded. This is repeated several times 
at the same mark, with the precaution of always moving the bubble up to the 
mark from the same side to eliminate differences of friction within the tube. Make 
a number of settings of the same end of the bubble at the mark just used, but by 
moving the end up to the mark in the opposite direction from that before used. 
Take the mean of this series of micrometer readings, and find the difference be- 
tween this mean and the preceding one for the same division. Compute the 
average deviation of each series, and determine if possible from these whether 
the difference between the means is purely accidental, or is probably clue to 
tardy action of the level. 

Make a similar series of settings from one side only for four successive 
divisions of the level, recording the numbers of the marks used, and whether 
towards A or B. Compute the means for each division. The difference 
between the means for successive divisions will give the difference of microm- 
eter reading corresponding to the interval between these two particular divisions 
of the level. From this compute the angle corresponding to each observed 
pair of successive lines. These will or will not be equal according to the 
uniformity of curvature of the surface of the tube. In the calculations note 
that the difference in micrometer readings divided by the measured distance 
AB, expressed in the same unit, gives the tangent of the angle through which 
the level is turned ; and that, as this angle is very small, its tangent coincides 
closely with its arc. Also, in any circle, the arc whose length is equal to the 
radius measures 206265". 



ESTIMATION OF TIME OF TEANSIT. 



ESTIMATION OP TIME OP TRANSIT. 

A weight having a large moment of inertia is suspended before a bright win- 
dow by a fine steel (piano) wire of such diameter and length that the time of a 
single torsional vibration is about half a minute. To this weight is secured a 
horizontal radial rod about half a meter long, carrying at its outer end, and at 
right angles to it, a vertical disk 2 or 3 decimeters in diameter, which is pierced 
near its centre by a small hole. The torsional vibrations of this apparatus, when 
once started by a slight twist, will decrease in amplitude so slowly as to con- 
tinue sufficiently large for observation during a quarter of an hour. A telescope 
with a vertical wire or with cross wires in its eye-piece, is placed at a distance 
of about 50 feet from the disk. The observation will correspond to the ordinary 
" eye and ear method" of observing the time of transit of a star as employed in 
astronomical observation. 

First adjust carefully the position of the cross wires or of the eye-lens so that 
the cross wires shall appear perfectly distinct. Then focus the telescope carefully 
on the disk, so that the spot of light and the cross wires of the telescope are both 
clearly seen at the same time. Point the telescope to about the middle of the 
path of the artificial star. The observation to be made is of the times of suc- 
cessive transit of the spot of light as it moves in the same direction across the 
vertical wire of the telescope. Start the metronome with its sliding weight set 
at 60, i.e., so that it shall beat seconds. Count the seconds continuously, 
observing through the telescope at the same time the posi- 
tion of the star at each stroke or second. The appearance 
will be somewhat as the sketch. Let the circle represent 
the field of view of the telescope, and the lines the cross- 
wires. The dots show the successive positions of the star 
at (say) 3, 4, 5, 6, 7, 8, etc., seconds. Then the star crossed 

J „ the vertical wire at some time between the 6th and 7th sec- 

Fig. 7. 

onds ; and as the wire is at about 0.4 of the way from the 

position at the 6th to that of the 7th second, and as the star moves nearly uni- 
formly at that part of its path, the transit must have taken place at 6.4 seconds. 
The counting is to be made continuously, and this having been done, the star 
will be found again approaching the wire in the same direction as before (the 
time of transit in reverse direction is not wanted) at (say) the 43rd second; and 
the position at 43 and 44 seconds having been mentally noted, as before, it 
would be estimated that the transit occurred at 43.8 seconds. These observa- 
tions are to be made continuously, recording at each transit the correspond- 
ing second and tenth of a second, until the time becomes 100 to 200 seconds, 
when a new series may be begun. In the counting, it will be found easier to 
count up to 50 and then drop the tens' place of figures, going on with 1, 2, 3, 
etc., and putting in afterward the necessary additional tens. 

Difficulty is frequently experienced in the first attempt at the observations, 
from the necessity of making the estimation, record, and count, at the same 
time. It is therefore sometimes best, in the first few observations, not to 
attempt to keep up the counting beyond the, first transit, but to make several 
disconnected observations until a little practice in the estimation is obtained. 

Every observation made should be recorded in the note-book. 

Since the rate of vibration of the pendulum is constant, the intervals between 
the times of successive transits should be equal. Find, therefore, the first dif- 
ferences and their average deviation from their mean. This gives a measure of 
the precision of the estimation. 



/3 t S e 


7 1 9 «\ 










\ (DIM 


*«*«« / 



METHOD OF WEIGHING. 



METHOD OF WEIGHING. 

The student unfamiliar with the use of a balance of even moderate delicacy 
should follow carefully the following suggestions and directions, so that he may 
avoid injury to the balance, and become methodical in making weighings, an 
essential to rapid and accurate work with the balance. 

The beam is supported by " stops" or bearings of one or another form, and 
it or the supports are movable by some mechanical device so that the beam may 
be made to rest'on its knife-edge bearing for weighing. This motion is usually 
made by turning a milled head, by raising or lowering a lever, or by some device 
apparent on inspecting the balance. The beam must be lowered very slowly 
and gently upon the knife-edges, as sudden blows or jars dull and spoil the 
edges and their supporting surfaces, and thus greatly injure the delicacy of the 
balance. No weights or substances, however small, should be placed upon the 
pans or taken from them ; nor should any operation whatever, involving a touch 
however light upon the balance or its case, be performed while the beam rests 
on the knife-edges. Weighing is a slow and delicate manipulation, and should 
be carefully and thoughtfully performed. 

Some balances are furnished with light " stops," which may be raised and 
lowered beneath the pans so as to just touch the bottom of the pans when the 
beam is horizontal. In weighing, these stops should be raised against the pans 
until the beam is lowered upon the knife-edges, and should then be slowly low- 
ered clear of the moving pans. They serve to lessen the blow upon the knife- 
edges, to stop the lateral swinging of the pans, which must never be allowed in 
weighing, and to lessen the strain upon the beam as the loads are being placed 
in the pans. 

When about to weigh, see first that the balance is level, turning the foot 
screws, if necessary, until the level, plum-bob, or other means of levelling, if 
any is provided, shows that the balance is level. See next that the beam is in 
its proper place — not twisted from its normal position on the stops by previous 
careless usage — and that the scale-pans hang properly from their knife-edges 
or supports. Lower the beam carefully, lower also the pan-stops slowly, and 
see whether the pointer or index moves equally on each side of the middle 
division of the scale at its foot, keeping the balance case shut to avoid disturb- 
ance by air currents in this and in the final parts of all weighings. If it does 
not, the balance is out of adjustment, and must be readjusted before weighings 
are made. For instruction upon this point, call upon the instructor. Immedi- 
ately upon the completion of every observation on the swing of the balance, 
before any other motion is made, the beam must be raised from the knife-edge, 
turning up first the pan-stops, if these are present. 

The substance to be weighed is placed in the left-hand pan, the weights in 
the other pan, on account of the greater safety and convenience of transferring 
the numerous weights from the box at the right hand to the pan nearest it. 
This practice should be habitual, as, in addition to its convenience, it eliminates 
the effect of unequal balance arms where only relative weights are desired. The 
error from unequal balance arms may be eliminated either by the method of 
double weighing, or by "tareing" (Kohlrausch, Phys. Measurements, p. 29; 
Thresh, ExpVl Physics, p. 23), or by measuring the ratio of the two arms as 
described in the notes on "Adjusting and Testing a Balance." All weights, 
except when very heavy and intended for rough work only, should invariably 
be handled by pincers. By moisture from the hand the weight may be 
increased by even a mgr., and rusting will ensue which will destroy the pre- 



8 METHOD OF WEIGHING. 

cision of the set of weights. For small weights, from 10 mgr. down, it is more 
convenient to use a "rider "than small metal weights. The rider is a bent 
piece of wire which may be moved along the graduated balance arm and hung 
upon it at any desired point, or removed by means of a carrying rod which pro- 
jects through the end of the balance case. 

In making the weighing, follow closely this manner of using the weights. 
See that the rider swings wholly clear of the beam. The substance being in the 
left pan, and the box of weights open upon the table (not on the balance case) 
in front of the right pan, transfer to the middle of the pan a weight which is 
thought to be somewhat larger than that required, — suppose* 50 gms. Lower 
the stops slowly — generally it is unnecessary, until the weights are nearly right, 
to lower completely — until the index moves to (say) the left, showing that 50 
is too large. Raise stops, and return the 50 to its place in the box (never place 
weights anywhere but in their places in the box or in the scale-pan) , and put on 
the next smaller, 20 gms. On lowering, index goes to right ; 20 is too small. Put 
on the second 20 (sometimes two twenties and one ten are provided, sometimes 
one twenty and two tens, — the former is preferable) ; too much. Eemove, and 
put on 10 ; too little. Put on 5 ; too much. Remove, and put on 2 ; too much. 
Remove, and put on 1 ; too little. Put on 0.5 ; too little. Put on 0.2; too little. 
Put on other 0.2; too much. Remove, and put on«0.1 ; too much. Remove, and 
put on 0.05; too little. Put on 0.02; too little. Put on other 0.02; too little. 
It is unnecessary to now put on 0.01, because that would make up 0.05 + 0.02 
+ 0.02 + 0.01 = 0.10, which has already been found too much. Add the remainder 
by the rider, keeping the balance case closed, finding such a point for it that the 
index will swing, as at the outset with no load, equal distances on each side 
of the middle point. If, in the final weighing, the balance does not readily 
swing of itself, a touch with the pan-stops or a very light touch with the 
pincers on one pan will suffice to give the motion of four or five divisions on 
each side. Then proceed to count up the weights missing from their places in 
the box, — for each weight should have a fixed place in the box, — and add to 
this sum the weight corresponding to the position of the rider (suppose 0.0024 
gms.). In this case the sum would be 31.7924 gms., and the figure in each deci- 
mal place should be put down as it is read off from the box.. Then open the 
balance-case and count up the weights in the pan, beginning with the largest, 
and noting, as each place of figures is completed, whether it checks with the 
already recorded number. Then read and remove the rider, and remove the 
weights in order, beginning with the largest, counting up the weights a third 
time, and noting whether this checks the previous numbers. By this repetition, 
which takes very little time more than is required for the mere manipulation, 
annoying mistakes in recording weights may be wholly avoided. 

On leaving the balance, always see carefully to the raising of the beam from 
the knife-edges, the removal of the rider from the beam, and the closing of the 
balance case, and of the box of weights, in which the pincers should be left. 

To familiarize himself with the above procedure, the student should make 
the weighiug to 0.1 mgr. of each of the two pieces of metal provided for the 
purpose, and then of the two together, as a check, recording the weights 
against the marks found upon the pieces. 



ADJUSTING AND TESTING A BALANCE. 



ADJUSTING AND TESTING A, BALANCE. 

A delicate balance should be properly protected from dust and excessive 
moisture. Drying substances within the balance case are sometimes, but not 
always, of service. They may maintain the air within the case at so low a humidity 
that sensible condensation or absorption of moisture by bodies being weighed 
does not occur. The balance should not be exposed to unequal or irregular 
heating, and should stand on a table as free as possible from jarring. Where a 
slight vibration is unavoidable, it may be lessened by placing masses of rubber 
beneath the feet of the case; but, as such a support is somewhat yielding, care 
must be taken to see that this does not introduce consequent errors into the 
weighings. In what follows, the balance is supposed to be so located, and to 
have been properly cleaned and put together. It remains to test and adjust it, 
and the following notes give the main points in this process. The student 
should note the number of the balance to which he is assigned; and should 
record against the number corresponding to the descriptive section the obser- 
vations and result obtained with the balance. Students unfamiliar with the 
use of a delicate analytical balance should perform the work indicated in the 
notes on " Method of Weighing" before going on with the work of these notes. 

1. Levelling. — Level the balance by either the plumb line behind the 
standard, the level on the base of the standard, or the level in the case, accord- 
ing to which is furnished with the balance. Great precision in levelling is not 
essential, but firmness of the balance case, so that this adjustment, once made, 
shall not be disturbed during the subsequent use of the instrument, is important. 

2. Equilibrium. — Release the beam from the stops; i.e., lower it by the 
method provided for the purpose until the kuife-edges rest upon the supporting 
plates. This motion and the reverse one must invariably be gentle, avoiding 
any sudden blows upon the knife-edges, or jarring of the beam or pans. The 
stops of the balance, when properly arranged, will release both sides of the 
beam at the same time. No changing of the load in the pans, no raising or 
lowering of the balance-case doors, — in short, no operation whatever on the 
balance should be performed, except in some cases by the experienced manipu- 
lator, while the beam rests upon the knife-edges. The pans must not be in 
vibration when weighings are made. When the beam is thus released, the index 
should swing freely and slowly equal distances on each side of the middle point 
of its scale. If it does not, see first that the pans are properly hung upon the 
beam, and then turn the adjusting nut until the index swings approximately 
equal distances on each side of the middle mark. A nut at one or each end of 
the beam, moving on a screw projecting from the end of the beam, or sometimes 
a small projecting piece or "flag" of metal revolving on a vertical axis at the 
middle of the beam, constitutes the attachment for this adjustment. If this ad- 
justment cannot be made, but the balance, on starting to one side or the other, 
continues to swing in that direction with increasing velocity, tending to over- 
turn, then it is in unstable equilibrium, and the large vertically moving adjusting 
nut at the middle of the beam (above or below) must be lowered until the proper 
equilibrium is attained. In the ordinary analytical balance, the time of a single 
vibration (swing from one extreme to the other, or, better, from the middle 
point to one extreme and return) should be from 5 to 15 seconds. 

Test whether the balance is in stable equilibrium, and note the time of the 
single vibration. This time can be regulated to the proper one for the purpose 
of the balance by raising or lowering the equilibrium nut. In any individual 
balance the sensitiveness increases with the time of vibration given to its beam. 



10 ADJUSTING AND TESTING A BALANCE. 

3. Resistance. — Excessive resistance to the swing of the beam, if clue to 
friction of moving parts, will be indicated by a failure to fulfil the two following 
tests, which should now be made. Note the position of the pointer on the scale 
at the extremes of several successive swings. The differences between succes- 
sive readings on the same side will show the diminution in amplitude due to 
friction and to resistance of the air. This should not exceed 0.1 to 0.3 of a 
division in a good analytical balance. Let the balance swing until it comes to 
rest, and read the position of the pointer; repeat several times. The successive 
positions of rest should not differ by more than 0.2 of a division.- Make each 
test with no load in the pans, and then with a moderate load (say 50 gms.) in 
each pan. 

4. Weighing by Swing of Balance. — In most other tests of a balance this 
method of weighing is necessary, and is therefore inserted here. There are two 
methods of weighing a body, of which the first, and ordinarily the most con- 
venient for moderate loads when the extreme limit of precision is not desired, 
is to first adjust the beam until the index swings exactly equal distances on each 
side of the middle division of the scale, allowing for the diminution of ampli- 
tude in successive swings ; then to place the body to be weighed in the left-hand 
pan and weights in the right, until, with proper adjustment of the rider, the 
index again swings equal distances on each side of the middle point. The 
second method is the more precise, and is more rapid with heavy loads on 
slow balances of precision, if not in all cases where really extreme precision 
is sought. It is applicable to balances without riders, and requires no weights 
smaller than 1 mgr. It is as follows : 

(a) Have the balances adjusted so that the index swings about equally on 
each side of the middle mark of the scale. Eincl the position of rest of the index, 
by Observing and recording the turning points, or extreme positions, of the 
index at both ends of several successive swings. For this purpose it is best to 
have the divisions of the scale numbered from right to left, calling the middle 
division 10, or some convenient number, since the weights are ordinarily used in 
the right-hand pan, and an increase of weight thus increases the index reading. 
This numbering is assumed in the examples below. Swings of not over 4 or 
5 divisions on each side are best, and an odd number of readings, say 5 (3 at 
one side and 2 at the other) or 7, should be taken to eliminate the effect of 
resistance. 

Example. — Load = 0. 



r. 


I 


6.2 
6.4 
6.5 


14.2 
14.0 


19.1 


28.2 


6.37 


14.10 




6.37 




20.47 


)int of rest : 


= 10.23 



This point of rest with no load will be varied by any disturbance of the level 
of the balance, and can only be used so long as it is known that the adjustment 
of the level is not disturbed. It must be redetermined frequently, and in general 
whenever any doubt could arise. 

6. Place in the left-hand pan the body to be weighed. Put weights into the 
other pan, and adjust the rider (or small weights, as the case may be), until a 



ADJUSTING AND TESTING A BALANCE. 11 

weight is obtained which is less than 1 mgr. too small, so that 1 mgr. additional 
would be too great. Suppose that in an actual case the smaller weight was 
6.154 gms., the larger 6.155 gms., and the turning-points of the index with each 
successively were as here recorded. 



wt.= 


6.154 gms. 


r. 




1. 


6.7 
6.8 
7.0 




13.2 
13.0 


20.5 




26.2 


6.83 




13.10 
6.83 
19.93 


Eest 


= 9.96 div. 



m.= 


6.155 gms, 


r. 


I. 


6.8 
7.0 

7.2 


16.7 
16.5 


21.0 


33.2 


7.00 


16.60 




7.00 




23.60 



Eest =11.80 div. 

.-. deflection for 1 mgr. under this load = 11.80 — 9.96 = 1.84 cliv. (This, as 
will be shown later, is the " sensitiveness " of the balance under this load, and 
may be determined from the curve or table of the sensitiveness, described in 
§ 5, with a considerable saving of time if many weighings are to be made with 
the same balance.) 

Now from the previous determination (a) , it is found at the time of this 
weighing that with no load the point of rest = 10.23. Hence the weight which 
would have exactly counterpoised the substance, i.e., which would have made 
the index again vibrate about this point of rest 10.23, would have been, 

Wt. desired = 6.154 gms. + 1023 ~ 9,96 mgr. 
h 1.84 & 

= 6.154 + 0.00014 gms. 

= 6.15414 or 6.1541, 

rejecting the last place of figures if, as usual, the weighings do not pretend to 
a precision of more than one-tenth of a mgr. Of course any other increment of 
weight than 1 mgr. may be used, but it is desirable that a small increment only 
should be taken, since for large differences of deflection the deflection may not 
be strictly proportional to the load. In general, let 

a = point of rest of index with too small weights, w gms. 
b= " " " " weights greater by m gms. 

c= " " " " no load. 



Then desired weight in gms., 



b — a 



By this method make a weighing of a piece of metal provided in the balance 
in this experiment. Record the mark upon the metal. All records of weigh- 
ings should be complete, and should be kept in regular form. 

5. Sensitiveness. — By the sensitiveness of a balance is meant the difference 
of index reading produced by the addition of 1 mgr. to one pan of the balance. 
If the beam were perfectly inflexible, and the three knife-edges in the same 
plane, the sensitiveness of the balance would be the same, whatever the load 
(supposed always equal in the two pans) on the balance. But as inflexibility is 
impossible, the sensitiveness of the balance must vary with the load, diminish- 
ing as the load increases. 



12 ADJUSTING AND TESTING A BALANCE. 

To study the sensitiveness of the balance, place equal weights in the two pans. 
Exact counterpoising is unnecessary. Determine as in (a) the point of rest a. 
Then add on one side (r supposed) , by means of the rider, a weight of 1 or 2 
mgrs. = m, and again find the point of rest b. Then the sensitiveness with this 
load (by " load " is always meant weight in one pan only) is 

b — a 
m 

Find the sensitiveness for each 10 gms., from up to 50 gms. (or up to the 
limit of the balance load, taking care not to injure the balance by excessive load- 
ing). Plot points with loads as abscissas and sensitiveness as ordinates, and 
draw a line through them. This curve or a table of the determined sensitiveness 
and corresponding loads may be used for interpolation in weighing, as suggested 
in (&), and may thus considerably facilitate work. 

6. Ratio of Arms of Balance. — In general, of course, there must be a greater 
or less difference between the length of the balance arms, i.e., of the distance 
of the outer knife-edges from the centre one. This distance is usually adjusted 
by the balance maker, so that the ratio of the balance arms shall be correct 
within the limit of precision for which the balance is intended ; and, in many 
cases where only relative weighings are desired, an error in this ratio is of no 
consequence. But, where accurate weighings are to be made, a test must always 
be had of the ratio of the arms. Call 

B = length of right arm of balance, 
L — length of left arm of balance, 

and suppose weighing always to be made with the weights in the right-hand pan. 
Then any weight w thus found to be sufficient to counterpoise the substance to 
be weighed must be multiplied by the ratio B^-L to reduce it to the true weight, 
which would be found if the balance were equal-armed. 

If two bodies or weights known to be precisely equal (within the limits of 
precision desired) were at hand, then the ratio could be found by placing one of 
these bodies in each pan, and then adding weights to one or the other pan until 
the pointer oscillated about the previously determined point of rest with no load. 
Suppose that a weight w r is necessary in the right pan ; then, by the laws of equi- 
librium of moments, 

(W+w r )B = WL; 
whence, 

R = W 

L W+Wr 

where W is the weight placed in each pan. If the extra weight required is in 
the left-hand pan, then w r = 0, and a corresponding term wi must be added to the 
numerator. 

But in general the nominally equal weights of a set are not, in fact, suffi- 
ciently nearly so for the present purpose, and the procedure must be as follows : — 
Let A and B be two weights or combinations of weights nominally equal, and, 
in fact, very nearly so ; and in amount, each about half the maximum load for 
which the balance is designed. Place A in the right pan, and B in the left. Add 
weights to the apparently lighter pan until the index vibrates equally about the 
observed " point of rest with no load." This small weight is best added by the 
rider. If various small weights are used instead of the rider, care must be taken 
that they are in proper adjustment, lest they introduce error. Suppose a small 
weight r to be required in the right pan. Then reverse A and B, and suppose a 
small weight I to be required in the left pan for equilibrium. If these small 
weights require to be in the reverse pans from those mentioned in either or both 



METEOROLOGICAL INSTRUMENTS. 13 

cases, it is simply necessary to account them negative weights in the pan above 
named. Then, by the law of moments, 

B(A+r)=LB and RB = L(A + l). 

Solving by eliminating B gives 

B* = A + l ) 

L 2 A + r 
which, by two approximations, since I and r are small referred to A, gives 

L 2 A L 2 A 

in which the observed values of I, r, and A may be substituted. 

In obtaining the small weights r and I necessary for the equilibrium, the 
method of using the swing of the balance, as described in (&), may be employed 
to advantage instead of equal oscillations about the point of rest. 
Example. — Maximum load of balance 100 gms. 

Right. Left. 

50 gms. (20 + 20 +10) + 0.0030 gms. 

(20+20 + 10) 50 + 0.0005 

... r =-0.0030 1 = + 0.0005. 

... g =1 + 0.0005+ 0-0030 = L000035 , 
L 2X50 

7. Errors of Set of Weights. — Weights from the best makers can seldom 
be relied upon as correct to 1 mgr., and, in precise work, it is therefore neces- 
sary to determine the relative errors of the weights of the set in terms of some 
one weight of the set; and further, if necessary, to find the absolute error of 
this weight by comparison with a standard weight. The method for this study of 
the weights may be found in Kohlrausch's Physical Measurements, p. 32, and 
in Thresh's Experimental Physics, p. 25, preferably the latter. Students will 
omit the work of this paragraph unless personally instructed to perform it. 



METEOROLOGICAL INSTRUMENTS. 

This experiment includes the reading of the Fortin and Siphon barometers, 
self-registering thermometers, the wet and dry bulb thermometers, the anemom- 
eter, etc.; and its object is to familiarize the student with these instruments. 

Barometers. — A description of the Fortin barometer may be found in Ganot's 
Physics, p. 122, and in Deschanel, p. 147. To read the barometer, turn down the 
screw at the bottom of the cistern until the mercury stands slightly below the 
ivory point projecting downwards from the top of the cistern. This insures a 
rising meniscus at the top of the column in the subsequent setting. Then turn 
the screw up until the ivory point just touches its image in the mercury surface 
when the barometer is stationary. If this position be passed, the point will pro- 
duce a depression in the mercury surface. Many observers prefer to turn up 
the mercury until a very minute depression is obtained, instead of observing 
simply the coincidence of the point and reflection. This method has some 
advantages, and is in general more accurate. When this is done, jar the tube 
near the top of the mercury column by tapping lightly upon the side with the 
hand. This tends to secure uniformity in the meniscus at different readings, 
eliminating the effects of unequal adhesion of the mercury to the walls of the 
tube. The vernier is arranged to slide up and clown by a rack-and-pinion move- 
ment connected with the milled head at the right-hand side of the tube. On the 



1 


4 /« 

1 CL 




1 


c ^ 





14 METEOROLOGICAL INSTRUMENTS. 

opposite side of the tube from the vernier is another similar metal plate which 
moves with the vernier, and has its lower edge in the same plane, perpendicular 
to the tube, as that of the vernier. If, therefore, the eye be raised or lowered 
until just in line with these edges, the line of sight will be perpendicular to the 
tube. This is very important. The setting is made by turning the vernier up 
until wholly above the mercury column, then turning it clown, the eye being 
kept as nearly as possible in the plane just described, until the edges, when in 
line, appear just tangent to the top of the meniscus, i.e., just cutting off light at 
the point a in the figure, while allowing it to pass through at the sides. While 
making this setting, the barometer should be turned toward a 
bright background. The vernier of the best English and Ameri- 
can barometers, having a scale of inches, usually reads to 0.002 of 
an inch, and by estimation to 0.001 of an inch. 

As the column of mercury at higher temperatures required to pro- 
duce the same pressure is greater than at lower ones, owing to the 
less density of the mercury, it becomes necessary to reduce the 
reading to a standard temperature. The thermometer on the front 
of the column has its bulb inside the brass tube, and gives quite accurately the 
temperature of the mercury; but when the changes of temperature are rapid, it 
becomes essential to surround the brass tube with a heavy wrapping of woollen 
material, to render changes in the air temperature less quickly felt by the mer- 
cury; otherwise, the whole column will not be at the same temperature, and the 
indication of the thermometer will be of uncertain precision. 

This precaution is necessary whenever an accuracy of 0.005 inch is sought. 
The Signal Service directions for reading the barometer require the reading of 
the " attached thermometer" before that of the vernier. The standard temper- 
ature adopted in scientific work and by the U.S. Signal Service, to which to 
reduce all observations is 0°C = 32 o F. As the barometer scale is usually of 
brass, this reduction may be made by dividing the reading at a temperature t by 
1 + 7 t, where y = the coefficient of apparent expansion of mercury on brass, 
= 0.000181 — 0.000019 = 0.000162 per degree Centigrade. The correction to be 
subtracted from the direct reading is, however, more conveniently found from 
tables such as are given at p. C, 63, or p. C, 79 of the " Smithsonian Collection." 
(The numbering referred to is at the bottom of the page.) An explanation of 

!the use of each table is given at its beginning. The figure shows the 
principle of the siphon barometer, and an inspection of the instrument 
will render the method of reading obvious. The graduation is num- 
bered each way from the centre, and the sum of the two readings thus 
gives the difference of level of the upper and lower meniscus. 

The third form of mercurial barometer among the meteorological 

instruments is known as the "Kew Standard." In this the area of 

the tube bears a known ratio to that of the cistern, and the scale is 

graduated so that each inch shall be shortened by that amount ; thus 

the rise of the mercury in the cistern is compensated without the 

adjustment necessary in the Fortin barometer; but the instrument, 

Fia " 9< although convenient, is less accurate than the Fortin. 

Read each of these instruments and the attached thermometers, and reduce 

each reading to 0° by the tables. Reduce the metric barometer reading to 

inches. (1 inch — 25.400 mm.) 

Calculate by Guyot's tables, " Smithsonian Collection," p. D, 33, the height 
of a place at which the barometer reading is 1.111 inches lower than the reading 
obtained by the Fortin, calling t = t = the observed temperature by the attached 
thermometer, t 1 = t' = 55.5°, and Lat. = 35°. 



METEOROLOGICAL INSTRUMENTS. 15 

Thermometers. — In Rutherford's maximum thermometer a small piece of 
iron may be brought by a magnet, or by shaking, into contact with the top of 
the mercury column, and will then be pushed forward as the mercury expands, 
but will be left behind by the retreating mercury column as the bulb cools, and 
its lower end will thus indicate the maximum temperature obtained. In the 
Rutherford minimum thermometer a glass index is placed within the alcohol 
thread. It is drawn back by adhesion and capillary action as the top of the 
thread retreats, and is left behind as the column advances, thus recording by 
its upper end the minimum temperature attained. The index is adjusted by 
tapping on the thermometer when in an inclined position. The Negretti and 
Zambra maximum thermometer is made with a contraction in the capillary near 
the bulb. When the mercury in the bulb contracts, separation takes place at 
that point, and the column remaining in the capillary indicates the maximum 
temperature. The reading should be taken with the tube slightly inclined 
downward toward the bulb, as an inspection of the instrument will show. By 
tapping the end of the instrument against the hand, or by a whirling motion, 
the thread may be forced back into the bulb. Read each thermometer, heating 
by the hand or breath if necessary to observe its action. 

Moisture of the Air. — The amount of vapor of water in the air may be 
determined by several methods, of which the direct methods of measuring the 
weight of some substance having a great affinity for water, before and after 
having been exposed to a known volume of moist air, are those upon which the 
others, more convenient in use, are based. The method in most common use is 
known as that of the " Wet and Dry Bulb Thermometer." 

The moisture of the air is usually expressed either by, 1st, the weight of 
water contained in the form of vapor in each unit volume; 2d, the ratio 
between the amount actually present to that which would be present if the air 
were saturated at the temperature of observation. The second is the more 
common, and the more generally useful method, and this ratio is called the 
"relative humidity" or " humidity" of the air. It is expressed in percentages. 
Thus relative humidity = 75 per cent or 0.75 denotes that the air contains three- 
fourths of the amount of moisture required to saturate it at the same tempera- 
ture. 

The use of the Wet and Dry Bulb Thermometer is based upon the fact that 
water at the temperature of the air evaporates less rapidly as the humidity is 
greater ; also that, in evaporating, the water absorbs heat from the bodies with 
which it is in contact. A glance at the apparatus will show its construction. 
By direct experiments it has been ascertained what difference of humidity 
corresponds to one degree difference between the two thermometers for any 
given temperature of the air. Upon these determinations have been based 
tables which are used in connection with the instrument. (See Ganot, 290-7, 
Deschanel, Stewart.) Observe upon the Negretti and Zambra Wet and Dry 
Bulb Thermometers the reading of each, taking pains not to warm either during 
observation. Turn to the " Psychrometrical Table," by Glaisher, Smithsonian 
Coll., p. B, 102, and by aid of the description of the table find the temperature 
of dew-point, force of vapor, weight of vapor, and relative humidity. Guyot's 
table, p. <B, 46, is probably more accurate, but does not exhibit these quantities 
quite as clearly. Compare these results with those taken from the tables of the 
U.S. Signal Service which are placed with the apparatus. The quite large 
discrepancies are due to the variations in the data upon which the tables are 
constructed. For greater convenience several graphical and other devices have 
been introduced into similar instruments to render the reference to tables 
unnecessary. One of these is called the " Hygrodeik." Set this instrument as 



16 COMPOSITION OF FORCES. 

directed upon the back of its plate, and read from it as there directed the four 
quantities above mentioned. Owing partly to errors in the thermometers, the 
results from this will not closely agree with those previously obtained. 

Anemometer. — Instruments for measuring the velocity of the wind are 
known as anemometers or air meters. One of the most perfect forms consists 
of a wheel carrying inclined fans, against which the wind impinges and causes 
rotation. This wheel is connected through others to hands moving" over a dial, 
upon which the velocity may be read off. A test of each instrument should be 
made, giving a correction to be applied to readings. A detent on the side of the 
instrument serves to throw the wheels in or out of gear. To make an observa- 
tion, throw the wheels out of gear, and take the reading of the hands. Hold the 
instrument in front of some ventilating register or in any air current. At the 
beginning of a minute again throw the wheels in gear, and at the end of the 
minute throw them out of gear. The difference from the former reading will 
give the velocity in feet per minute. Calculate the velocity in miles per hour, 
which is the time usually employed as unit in meteorology. 

Of the other forms of anemometers in use, the "cup anemometer " (used by 

£ U.S. Signal Service) is the most common. At the end of 

J | each of four arms at right angles to each other are placed 

H ^ hemispherical cups, as shown in horizontal section in the 

s-/ \~~ ~~) figure. The wind coming in the direction of the arrows will 

meet with less resistance from the convex surface than from 

j the concave one, and rotation will be set up in the direction 

Fig. 10. . ., , 

of the curved arrow. 



COMPOSITION OP FORCES. 

Place the blocks to which the spring balances are attached at any convenient 
positions on the hoop. With the weight removed from the vertical cord, and 
with the cords wholly slack, read each balance. This reading will be the index 
error of the balance, and must be added to or subtracted from all subsequent 
readings with the balance in that position, according to whether it is negative 
or positive. This index error must be determined for each position of the 
balances. 

Attach the 5-lb. or 10-lb. weight to the third cord. Turn the adjusting 
screws at the top of the balances until the cords meet at the pin passing through 
the centre of the circle. Read the spring balances, and record these and all 
other observations at suitable places on a sketch showing the direction of the 
cords. Find the exact direction of the three cords by moving the fine thread 
attached to the pin in the centre of the graduated circle until it is parallel to 
each cord in turn, taking the corresponding reading on the circle to tenths of a 
degree. 

Considering either pair of the three forces acting, then the other must be 
equal in magnitude and opposite in direction to the resultant of this pair. 

Make the necessary observations with three or more positions of the 
balances. Find for one set of these, by geometrical construction to scale, the 
resultant of two forces, only one of which is a balance reading. 

From the triangle of forces it follows, that when three forces are in equi- 
librium about a point, these forces are to each other as the sines of their 
opposite angles in the triangle. Therefore, in any one set of the observed 
results, the ratio of each force to the sine of its opposite angle should be the 
same. Compute the value of each ratio in all the cases tried. 



DIVIDING ENGINE. — CBANK MOTION. 17 



DIVIDING ENGINE. 

Paragraphs 1, 2, 3, and 4 of experiment 21, as described in the " Physical 
Manipulation," p. 56, are to be performed. 

1st. To measure the back lash. Focus the microscope carefully for any very 
fine and distinct point upon the glass plate. Any mimute scratch or irregularity 
of surface is suitable. Bring the cross hairs up to this point from the right-hand 
side, taking pains that the motion of the screw is always in one direction, so that 
the nut is bearing up on one side against the threads of the screw. Kead the 
position of the micrometer. Then turn the screw until the cross hairs are upon 
the left-hand side of the point, and bring the cross hairs then up to the point 
from this (left) side of it, taking pains as before that the screw is turned in one 
direction only, so that the nut is now bearing against the screw thread in the 
opposite direction from that in the first setting. Eead the micrometer. The 
difference of these two readings will give the extreme amount of play of the nut 
upon the screw at this part of the screw; that is, will give the "backlash." 
Take a number of settings in each direction, and compute the " Average Devia- 
tion " of the setting. 

2d. For measuring the pitch of the screw, use the steel scale of inches pro- 
vided with the apparatus, and find how many revolutions of the micrometer, 
reading always to 0.001 revolution, correspond to an inch upon the scale. Let 

this be n revolutions. Then the pitch of the screw is - of an inch. The meas- 

n 

urement thus made is, of course, not of a high degree of precision, owing to the 
kind of scale used, and no temperature correction need therefore be applied. 

3d. Measure the distance between two successive pairs of crosses upon the 
glass plate, recording the number of the crosses used. 

4th. Measure the two co-ordinates of a series of points (not less than seven) 
on the larger smooth curve on the plate of acoustic curves and plot the curve. 



CRANK MOTION. 

These notes supplement the description at p. 68, "Physical Manipulation.' 
First measure AD, as may readily be done with sufficient 
accuracy without removing it from its place. Then meas- 
ure CD by reading A with F first at 0° and then at 180°. 
The zero-point 0° should be in line between C and A. To 
test whether it is so, and to find the error, set inexactly at 
90°. Read A. Turn F until A reads again the same, Fl6, u " 

when F will have come to, or nearly to 270°. Then 270° minus the actual reading 
of F will be twice the index error of the circle, which must be allowed for in 
setting F in line with C and A, and in all the subsequent settings at 10°, 20°, etc. 
The explanation of the last correction will be seen by ^^>, ' 
inspection of the exaggerated sketch in Fig. 12. Only /\ s r--\^_ 
one length of -arm CD is to be used in the experiment. [' gL.-X-777^>£!L- 

In computiug "the distance of AB from the mean posi- V A fx---*~*"" 
tion," CD must of course be expressed in the same units ^LS^ 
as those in which the reading at A is taken. Also it is to Fig. 12. 

be noticed that the "mean position" referred to is the mean between the read- 
ing of A when F is 0° and when F is 180°. The observed distances of A from 
this mean position are obtained by subtracting from the observed readings of 
A this mean position reading. 




18 ANGLE OF FRICTION. — BREAKING WEIGHT. 



ANGLE OP FRICTION. 

Read pp. 71, 72, " Physical Manipulation." 

In the apparatus used in the laboratory, the distance from the hinge F to the 
left-hand edge of the scale rulings is one metre, and the scale is numbered in 
decimals of a metre, so that the readings where the wire index crosses this edge 
give tangents of the angle DAB directly. Now the weight of the body E may be 
represented by the vertical line om, Fig. 13 ; this force may 
be decomposed into the two others, on, perpendicular to 
AD, and mn, parallel to it. The former, on, produces pres- 
FiQ 13 sure on AD ; the second, nm, tends only to make E slide 

along AD. The coefficient of friction of motion is the ratio 
of mn to on, when the angle DAB is just sufficiently large to have E continue in 

uniform slow motion when once started. But the ratio — is the tangent of 

on 
mon, and mon = DAB ; hence the coefficient of friction is tan DAB, which, as 
before stated, is given directly by the scale reading. In the experiment, take 
five readings for each of the three substances furnished, determining the coeffi- 
cient of motion only, and take the mean of each series. It will be noticed that 
with the same substance the block will move with a different velocity, according 
to the angle used. Take pains, therefore, that the readings are always taken 
when the block moves with the slowest attainable velocity. Always start the 
block from the reference mark across AD, and let it slide a short distance only 
so that it may not change to an entirely new part of the surface. Set the block 
in motion by tapping it gently. 



BREAKING WEIGHT. 

With this apparatus the breaking weight, or the stress required to break 
wires of iron, copper, etc., is to be directly measured, and from this and the 
diameter of the wires broken, the ' ; tenacity " is to be calculated. The " tenac- 
ity," also known as the " ultimate strength," " modulus of rupture," and " resis- 
tance in tearing," is usually expressed by the weight which is just sufficient 
to break a rod of unit cross-section when applied in the direction of the length 
of the rod. If the weight be expressed in pounds, and the area of cross-section 
in square inches, then the tenacity would be in pounds per square inch; that is, 
would be the number of pounds weight which, when applied longitudinally, 
would be just sufficient to tear apart a bar of 1 sq. in. cross-section. This quan- 
tity may of course be also expressed in grammes per sq. centimeter, or kilo- 
grammes per sq. millimeter (a commonly employed unit), or, following the abso- 
lute system, in dynes per sq. centimeter. 

Cut off a piece of wire of the length indicated by the marks on the front of 

the apparatus. Twist one end of this piece 
'• c / ( ) "H around one of the pins in the crank at the 

,~ / U f; right-hand end of the frame, first raising the 

Ag yN wedge C (Fig. 14) nearly out of the frame, so 

i > OT ^^ >S that the balance shall be strained to but one 

FlG> 14i or two pounds. Pass the wire around the hook 

at B, so that it shall lie in the groove cut for it, 

and then attach the other end to the second pin in A. In both the attachments 

at A the wire should be twisted around the pin on the lower side, and pass 

between the upper projections. By this arrangement the wire is not sharply 



DEFLECTION OF BEAMS. I. 19 

bent at any point where the greatest strain occurs, and is therefore not weak- 
ened by the method of attachment. After the wire is fastened in place, turn the 
crank slowly so as to wind the wire upon it, and see that the wedge slides freely 
down as the balance spriug is stretched. When the wire breaks, the wedge will 
retain the balance at the reading at which it stood when the wire broke, and this 
reading will be twice the breaking weight, since the wire is double and each 
strand carries but half the total pull. Record in one column the diameter of the 
wire before breaking, as measured to ten thousandths of an inch by the microm- 
eter gauge, in the next column the diameter near the point of rupture after break- 
ing, and in the third the observed balance readings. Break three samples of 
each kind of wire provided. Calculate from the mean of each set of three the 
tenacity of that wire. 

Let d = diam. after breaking, 

w = weight required to break a single strand ; then 

ird 2 

—7- = area of cross-section of wire, and 

— -. — —~ y = tenacity = wt. req. to break a wire of unit cross-section. 

4 
Rankine gives for wires of 

Iron, tenacity = 70,000 to 100,000 lbs. per sq. in. 
Copper, tenacity = 60,000 lbs. per sq. in. 
Brass, tenacity = 49,000 lbs. per sq. in. 

Remembering that 

1 lb. = 453.6 grammes, 
1 sq. in. = 6.452 sq. cm., 

calculate the tenacity of the iron wire in grammes per sq. cm., in kilogrammes 
per sq. mm., and in dynes per sq. cm. 



DEFLECTION OP BEAMS. I. 

Supplementary to pages 77, 78 of " Physical Manipulation." 

In this experiment make readings as follows : — 

1st ; page 77, line 2 from below. Find deflection for every hundred grammes 
from to 1 kgr., carefully remove the load from the pan, and take the readings 
in each case with no load as the zero reading for that load ; and in the calcula- 
tions use for each deflection its corresponding zero reading. Be careful to turn 
the micrometer screw up before removing the load, otherwise the platinum con- 
tact surface upon the beam will become indented and injured. 

2d; page 78, line 7. Perform in full to line 21. Substitute for the remainder 
of the paragraph the following, and omit the last paragraph on page 78. 
* The deflection a of the centre of the beam is 

Wl 3 
4£&#' 
where b = breadth of beam = 6.73 mm., 
d = depth of beam = 3.73 mm., 
I = length of beam between supports, 
W= weight applied at middle of beam, and 

E = modulus of elasticity of the material = the ratio between the weight 
applied longitudinally to a rod of unit cross-section and the elongation thereby 
produced. 



20 DEFLECTION OF BEAMS. II. 

Since all these quantities are known except E, this may be calculated. Sub- 
stitute the observed value of a when W— 1 kgr., and compute E. In computa- 
tions, all dimensions must be expressed in terms of one unit. If this be the 
millimetre, E will be in kgr. per sq. mm. Calculate E in this unit; then, noting 
that 1 kgr. = 2.205 lbs., and 1 sq. mm. = 0.00155 sq. in., calculate E in lbs. per sq. 
in. Eankine gives in the latter unit, for steel bars, £=29,000,000 to 42,000,000. 
The steel (Stubb's steel) used in this experiment has a lower value than this. 



DEFLECTION OP BEAMS. II. 

In a beam of uniform cross-section, imbedded at one end and loaded at the 

other, the greatest angular deflection is given by the expression i = - • tt^-> 

2 El 

where 31= moment of load referred to supported end A of beam, 

I = length AB, 

E = modulus of elasticity of the material of the beam, and 

J== moment of inertia of its cross-section. 

This formula is derived on the hypothesis that the deflection is so small that the 
arc is sensibly equal to its tangent, and is, of course, only true within these 
limits. Using the gravitation system of measuring forces, M = Wl, E = the ratio 
between the weight applied longitudinally to a rod of unit cross-section and the 
elongation thereby produced, / for a rectangular cross-section = ^ of the prod- 
uct of the breadth b into the cube of the depth d of the section ; 



WK. 




\J 



Ebd* 

Similarly the vertical deflection of the end B of the beam will be 

= 1 . W£ = 4 . Wf_ m 
3 EI Ebd*' 

In the apparatus used, the telescope is 
to be placed opposite the mirror, and so 
adjusted that the reflection of the scale is 
plainly seen, and that OB from the scale 
to the mirror is about one metre. If the 
15 * beam be then loaded at any point, as C, 

the mirror will be tipped forward until its new position makes with its former 
position an angle i, which, since the portion CB of the beam remains unbent, is 
evidently the same as that which the surface of the beam at C makes with its 

* 0N~ 

original position. The angle OBN will be twice this angle; hence — =tan2£, 

OB 
the line of sight OB being always in the same direction. Now, as i is neces- 
sarily limited to very small values, the arc is sensibly equal to the tangent ; hence 

. _ tan2i _ ON , , 

2 ~2 0B k ) 

Take the scale reading when no load is applied. This gives the zero error 
by which subsequent readings are to be corrected. Place the weight of 1 kgr. 
at the point marked 50 cm., and read the point where the intersection of the 
cross-hairs appears on the scale. Put on 2 kgr. at the same point, and read the 
deflection. The angle i should be twice the former, i.e., the scale readings 



LAW OF PENDULUM. 21 

should be as 1 : 2. Calculate their ratio. Eepeat at the 30 cm. point. Then place 
1 kgr. successively at the 10, 20, 30, 40, and 50 cm. points, and take the corre- 
sponding scale readings ON, observing first in each case the reading with no 
load. From (1) it appears that the angular deflection should be proportional to 
the square of the length of the beam, i.e., of the distance AC; hence, by (2), if a 
series of points be plotted with abscissas proportional to the squares of 10, 20, 
etc., and ordinates to values of i; or, since i is directly proportional to ON, 
with ordinates representing the successive values of ON, these points should 
lie along a straight line. A slight variation from a straight line may arise from 
imperfect rigidity in the supporting pieces at A. 

From the data thus obtained, the modulus E may be calculated, since in (1) 
all the remaining quantities are known, assuming b =0.83 cm. and d = 0.818 cm. 
Insert then in (1) the values of arc i, W, Z, b, and d, when I = 50 cm. (using same 
unit of length in all cases), and calculate E. The ordinary value of E is about 
20,000 kgr. per sq. mm., but the steel (Stubb's steel) of the beam used here has 
a lower modulus. 



LAW OP PENDULUM. 

Give to the string which suspends the ball such a length that the ball shall 
hang near the floor. Measure the distance from the point of suspension to the 
top of the ball, expressing the result in metres and decimals. Add to this half 
the diameter of the ball, obtaining this by calipers. The sum of these would 
give nearly the length of equivalent, simple pendulum, but the centre of oscilla- 
tion is, in fact, slightly below the centre of the ball, and the actual length 

2 r' 2 

desired will be found by adding to this sum , where r= radius of the 

5 h + r 

ball. Set the ball swinging through an arc of 10° or 15° only, and find time 

required to make 50 or 100 siugle vibrations. Deduce from this the time of a 

single vibration. Eepeat with several lengths of pendulum, the shortest being 

not much less than a quarter of a metre. 

If T is the time of a single oscillation, and L the length of the pendulum, 

then T=mL n . Find, from observations just made, the values of m and n 

by application of the " logarithmic method." The complete equation would be 

T=ir-J—; whence it appears that the observations should give n = - and 

__jr_ 

n ~ /-. From the value of m found from the curve, calculate a. The metre is 

y/g 

used as the unit of length in order to bring the axis of Y among the observed 
points, so that m may be accurately determined. If any other unit were used, 
the intercept would lie considerably beyond the observed points, as will be seen 
on inspection. Students who have not studied the graphical method will com- 
pute g from each length of pendulum used by the formula t = rr 'y— 

y 



METRONOME PENDULUM. 

To prove the law of the simple pendulum by means of the metronome pendu- 
lum, we make the latter of such form that from the known dimensions and 
weights of its parts we can calculate the length I of the equivalent simple pen- 
dulum, i.e., of the simple pendulum that would oscillate in the same time as the 



22 METRONOME PENDULUM. 

metronome pendulum used. Then, by comparing I and t, we may find the rela- 
tion between them. Suppose the pendulum to consist of movable weights, w x 
and w 2 , sliding on the rod AB. Let the distance of the centre of 
gravity of each from the knife-edge C be fixed for an experiment 
at l x and l 2 respectively, and the rod be considered without weight. 
Then the length of the equivalent simple pendulum would be 

~ w x l l + w % l 2 ' 
Note that l 2 , being measured downwards, is always negative, so 
that the denominator, although an algebraic sum, is an arithmeti- 
■ FlG * 16- cal difference. To correct this result for the weight of the rod, 

(r 2_1_ r 2 a 2\ 
1 2 ^ — , where 

TT=the weight of the rod, a its half length, and r x and r 2 its inner and outer 
radii respectively. Thus we have 



H5M) 



(2) 

w x l x + w 2 1 2 

In this expression the numerator is the sum of the moments of inertia of the 
three parts of the pendulum, and the denominator is the sum of the statical 
moments, that of the rod being zero, since C is adjusted to pass through the 
centre of gravity of the rod. That the distance from the centre of support to 
the centre of oscillation is equal to the sum of the moments of inertia, divided 
by the sum of the statical moments, is here assumed. The demonstration 
may be found in treatises on mechanics. See Rankine's " Applied Mechanics," 
pages 516-518, and the Physics lecture notes. 

For the experiment, first place both weights near the ends of the rod, taking 
the pendulum off the supports while adjusting the weights. Observe the time 
of 100 single vibrations. In making these observations, the pendulum should 
swing through a very small arc, say 10° or 15° only, and the time of vibration 
should be measured as the pendulum crosses its position of rest, not from the 
end of the vibration, since the slowness of the motion there renders the result 
less precise. Then lay the pendulum upon a table, and with a scale of milli- 
meters measure to tenths of a mm, the distance from the knife-edge upon 
which the pendulum was supported to the nearest surface of each weight. Add 
to each half the thickness of the weight, and the results will be l x and l v Repeat 
this observation with w 1 about two- thirds of the way from the end of the rod 
to C, and again with the lower weight also part way up to C. Deduce for each 
case by means of equation (2) the corresponding value of I. Substitute these 

values successively in the expression $=»-%/-, which gives the law of the simple 

pendulum, and compare the computed times thus obtained with the observed 
times. They should agree within about one per cent. Errors in computation 
will be more likely to be detected, and time will be saved by arranging the 
calculations and results in a systematic form. 

The following data are given, but r 2 and the thickness of the weights are to 
be measured with calipers. 

.^ = 355.1 gms. 2a= 80.20 cm. 

w 2 = 383.3 " r x = 0.30 " 

W= 300.0 " # = 980.4 " 

In the result I will be found negative. This denotes simply that the equivalent 
simple pendulum would be one whose length would be below the point of 
support. 



bokda's pendulum. 23 



BORDA'S PENDULUM. 

Eead " Physical Manipulation," pp. 85-87. 

Determine first the time of vibration of the pendulum as there described ; 
then remove the pendulum carefully from its support, and lay it in the frame 
provided for it. Take it to some light window, and measure its length as 
follows : Focus one of the reading microscopes upon the knife-edge, and the 
other upon the rod, adjusting their positions until the intersection of the cross 
hairs is at the knife-edge in one, and at the point where the rod enters the upper 
side of the ball in the other. A metre scale lying on the portion of the frame 
marked A serves to measure the distance between the cross-wire intersections 
in this way. The frame is moved toward the microscopes until the scale is 
visible beneath one of the microscopes, and points toward the other. It is too 
short to reach it. The scale is further adjusted until some division, a, is under 
the cross-wires. This microscope is now removed without disturbing the scale, 
and set down at some point further along the scale of which the reading is b. 
Without disturbing the microscopes, the scale is now moved lengthwise until it is 
clearly seen under both microscopes, the readings being taken as c and d 
respectively. Then (a — b) + (c — d) = h will be the desired distance from the 
knife-edge to the top of the ball. Or three microscopes may be used. The 
diameter of the ball is now to be carefully determined by making several 
measurements, by means of calipers, of the vertical diameter. Call oue-half this 
distance r. All these measurements are to be made as nearly as possible to 
tenths of a mm. The distance from the knife-edge to the centre of oscillation 
of the pendulum is then 

5 h + r 

From this compute the value of g, by means of the formula for the time of 
vibration of a pendulum. As explained in the notes for the metronome 
pendulum, I is the moment of inertia divided by the statical moment, and the 
above expression may then be deduced. See Rankine's " App. Mech.", pp. 516, 
517, and 518, and the Physics lecture notes. 

In the apparatus as arranged, the error which would arise from the moment of 
inertia of the rod carrying the ball has been eliminated by adjusting the distance 
of the small ball above the knife-edge to such a point that the system consisting 
of this ball and the long rod vibrates in as nearly as possible the same time as it 
is intended the pendulum shall vibrate when the large ball is in place. This ball 
is then put on and adjusted to the proper position by trial. 

As the rate of the clock pendulum is liable to variation from day to day, its 
rate must be determined from a ten-minute observation, finding exactly the 
number of minutes and seconds by the dial corresponding to just ten minutes by 
the watch. Let m be the observed number of seconds by the watch equal to n 
by the seconds pendulum. Then one beat of the supposed seconds pendulum 

would be — seconds. After having computed, as described in the text-book, 

n 
the rate of vibration of the Borda's pendulum, that rate must be multiplied by 

this ratio, — , to reduce it to true seconds. The value of the fraction — must be 

n n 

computed to the fourth place of significant figures in order to correspond in 
accuracy with the remainder of the work, and more precise rating than that pro- 
posed would be preferable. § 



24 TORSION PENDULUM. — VOLUME OF FLASK. 



TORSION PENDULUM. 

A heavy cylinder of metal is supported by a thin steel wire, which, by its 
elasticity, can maintain torsional vibration in itself and the cylinder for a con- 
siderable time. The cylinder carries a pointer which moves over, a graduated 
circle, and serves to measure the amplitude of the torsional swing of the 
pendulum. A clamp screw moving along a vertical rod near the wire can be 
screwed to the rod and to the wire, thus fixing the length of .the wire, that 
portion only being under torsion which is between the clamp screw and the 
point Avhere the wire becomes rigidly attached to the cylinder. 

The time of vibration of the pendulum is independent of the amplitude of the 
vibration. Twist the cylinder, so that when swinging the pointer shall move 
through an angle of about 45°, 90°, 180°, and 360° successively. Note in each 
case the time of 50 or 100 single vibrations. The time should be very nearly 
the same in all cases with the same length of wire. 

The time of vibration is proportional to the square root of the length of the 
wire under torsion. Give the wire successively four different lengths, varying 
from the longest attainable to about one quarter of that length. Measure in 
each case the length of wire under torsion, and the time of 100 single vibrations. 
Since the time is proportional to the square root of the lengths, the quotient 

t 
—-= should be the same in all four cases. Compute and compare this ratio. State 

the extreme error in percentage. 

In timing vibrations, the times should be taken as the pointer passes its 
position of rest, not from the turning point, since the slowness of motion at the 
latter point renders the observation less precise. 



VOLUME OP FLASK. 

I. To determine the volume of the glass flask. Weigh the. flask empty. Fill 
it with water so that the bottom of the meniscus is just opposite the reference 
mark on the neck. Weigh again, and subtract. The difference gives the ap- 
parent weight w of the water in air. Find by a thermometer the temperature t of 
the water at the time of weighing. Also, read the barometer, and call H the read- 
ing in inches. The water as weighed in the air is buoyed up by an amount equal 
to the weight of air displaced. This weight is proportional to the volume and 
to the atmospheric pressure H. The volume of w grammes of water is, nearly 
enough for this correction, w cubic centimeters. Now one cubic centimeter of 
air at the ordinary temperature of the room, and under a pressure of 30 inches, 
weighs 0.00128 gms. Hence the weight of air displaced by the water will be 

TT 

0.00128 x wx— • Tlie brass weights are, however, buoyed up in the same way; 
but, since their specific gravity is about 8, their volume is only f- that of the 

TT 

water. Hence, the total correction for buoyancy will be only | X 0.00128 X w X — 

30 
gms. This added to vo will give the weight P of the water in vacuo ; and hence, 

p 
from the definition of the gramme, the volume of the flask will be V— — cu. cm., 

where D is the density of water at the temperature t, as given in the tables at 
the end of these notes. All weighing should be performed by Borda's method 
of double weighing. See p. 19, " Physical Manipulation." 



SPECIFIC GRAVITY OF SOLIDS. 25 

II. Determine similarly the volume of the small bottle, but use mercury 

P 
instead of water. As before, V— — . Let D= 13.596 = density of mercury at 

7)0 

0° C. ; then its density D at t° C, will be D = . In this case the weight 

J 1 + . 000182 1 fe 

of the air displaced by the brass weight is greater than that displaced by the 
mercury, so that the apparent weight of the latter is too great. Calculate the 
proper correction for buoyancy, assuming the sp. gr. of the brass as 8 and of 
the mercury as 13.6. Calculate also what would be the volume of the bottle at 
0° C, assuming .000026 as the coefficient of expansion of glass per degree Centi- 
grade. 

In filling the bottle with mercury, fill it first completely, then force the 
stopper firmly into place, allowing the overflow to run into the dish. Be careful 
before weighing that no globules remain attached to any part of the bottle, 
especially around the edge of the stopper. In using mercury avoid bringing 
into contact with gold finger-rings, etc., as they become amalgamated and 
injured. 

Density of Water. 
t D t D. 

0° 0.99987 16° 0.99899 

4° 1.00000 18° 0.99861 

8° 0.99988 20° 0.99821 

14° 0.99929 22° 0.99778 



SPECIFIC GRAVITY OP SOLIDS. 

The specific gravity of a substance (solid or liquid) is its weight in vacuo 
divided by the weight in vacuo of an equal bulk of distilled water taken at its 
maximum density, i.e., at 4° C. As the substance whose density is to be deter- 
mined varies in density with change of temperature, its specific gravity should 
be referred to some standard temperature. For this, either 0° C. or 62° F. 
(= 16.7° C.) are ordinarily selected, the former being in general preferable when 
the coefficient of expansion of the substance is well known; but when this is 
not the case, the latter (or frequently 60° F. = 15.6° C.) is better, since this 
temperature correction is thus rendered much smaller, and may often be wholly 
neglected. The standard temperature used should always be specified in stating 
results. It is often sufficient, however, to omit the reduction, merely stating 
the temperature at which the determination was made. 

In the determination of specific gravity of a solid, four weighings and two 
temperature measurements are necessary. 

W= weight to mgs. of bottle filled with water at temperature t°. 
Wy= " " " " + solid and water at temperature t x °. 

S= " " " solid dry in air. 

6= " " " bottle. 

D = density of water at t°. ) _ _ . _ . . .. ■ . ' ._ 

n _ u t tt | To be taken from tables at the end of these notes. 

&=mean coefficient of cubical expansion of glass, assumed to be the same at 
all temperatures, and approximately = 0.000026 per 1° C. 

Computation of Weight at t°. — As the temperature t° in general is not the 
same as t^, it becomes necessary to compute from W what this weight would 
have been had these been the same. Call this computed weight a v Assume, for 



26 SPECIFIC GRAVITY OF SOLIDS. 

( higher ) 
clearness, that t x ° is j & \ than t°. Then the capacity of the bottle on being 

( ITlf* V(*f\ CiPfl } 

heated from t° to «,° would be J _. . . _ , [ by its thermal expansion in the pro- 

( diminished J 

portion of 1 : 1 + h (i^ — t) when t and t x are not widely different. The exact 

1 4- kt 
expression is - — ~- The weight of water contained in this increased volume, 
1 + kt 

if the density remained the same, would therefore be 

(wv&)o+*0i~o:i5 . a) 

*.- * - * j ^ i ( decreases ) . ( heated ) 

but, in fact, the density ] . > from D to Z>, when the water is \ . . \ 

' J ( increases ) * ( cooled j 

from i to t v Hence, the weight of the water at t x ° would have been 

( T7- 6) [l + fc^-O]^. (2) 

and of the bottle and contents, this plus the weight of the bottle, i.e., 

<h = (W- V) CI + * (*i - *)] % + 6- (3) 

Reduction of Weighings to Vacuo. — The apparent weight of a substance 
(in air) is less than its true weight (in vacuo) by the weight of a volume of air 
which is equal to its own volume less the volume of the weights used. 

Corrections for this loss of weight are required to the apparent weight of 
the solid (— S), and to the apparent weight of the equal bulk of water (= w x ) 
only, since all the other apparent weights cancel, and the error is thus elimi- 
nated. This apparent weight w v in grammes, is 

Wi = 0i-(TT 1 -5). (4) 

Let A be the weight of 1 cc. of air at the temperature and barometric pres- 
sure in the room at the time of the weighing. These must be observed when 
PFand W l are taken; and in the following correction all weighings are supposed 
to be made under the same conditions of pressure and temperature, as otherwise 
the correction becomes somewhat less simple. Also, the humidity of the air is 
neglected, as of slight effect. Tables of the weight of 1 cc. of air at various 
pressures and temperatures are given at page 27. 

The volume of the "equal bulk of water" weighing w x grammes is (nearly 
enough for present purposes) w x cc; that of the counterpoising (brass) weights 

is, therefore, ^ 1 . The volume of the solid is, of course, w x cc., and that of the 
8 *5 

counterpoising weights is — . 

8-5 

Then, calling the specific gravity of brass 8.5, the loss of weight of the 
water owing to the buoyancy of the air is (nearly enough for the purposes of 
this correction) 



that of the solid is 



|)^=f 1 -l-, 



Now, adding (5) to w x gives w, the weight in vacuo of the bulk of water at 
t°, equal to the volume of the solid at the same temperature; and adding (6) 
to S gives the weight, which will be called s, of the solid in vacuo. 



SPECIFIC GRAVITY OF SOLIDS. 



27 



Specific Gravity referred to Water at fc°. — The ratio - would now give 

w 

the specific gravity of the substance at the observed temperature t x °, referred 
to water at the same temperature. But specific gravities are properly referred 
only to water at its temperature of maximum density, viz. 4° C. ; hence the 

Reduction to Water at <4° C. — Water weighing w grammes at ^° C, and 
having the density D v would have a volume — cc. at *,°; and, since at 4° C. its 
density is unity, this volume, i.e. this number of cc. of water at 4°, would weigh 



A 



grammes. Hence, the desired 
Specific Gravity of Solid at t x °, referred to Water at 4°, 



will be 



sp. gr. = s+ — 



sD, 



(7) 



If now it is desired to give the sp. gr. referred to some standard tempera- 
ture t 2 Q (any desired temperature, e.g. 0° C.)> it is necessary to know the mean 
coefficient of thermal expansion of the solid from 0° to t x ° and from 0° to t 2 °. 
Call these £ x and £ 2 , respectively ; then the desired Sp. Gr. at t 2 ° is given by the 
proportion 

Sp. Gr. = 1 + flA 
sp. gr. l + )8 2 £ 2 ' 



(8) 



the specific gravities being inversely as the volumes of the same mass. Ordi- 
narily & x and j3 2 may be assumed to be the same, but it should be observed that 
a definite and determinable precision is necessary in j8, in order that a greater 
error be not introduced in this last reduction than in the previous parts of the 
actual determination. 

Weight of 1 Cubic Centimeter of Dry Air. 





730 mm. 


740 mm. 


750 mm. 


760 mm. 


770 mm. 




grms. 


grms. 


grms. 


grms. 


grms. 


10° c 


0.00120 


0.001215 


0.00123 


0.00125 


0.001265 


12° C 


0.00119 


0.001205 


0.00122 


0.00124 


0.001255 


14° C 


0.00118 


0.001195 


0.001215 


0.00123 


0.001245 


16° C 


0.001175 


0.00119 


0.001205 


0.00122 


0.001235 


18° C 


0.001165 


0.00118 


0.001195 


0.001215 


0.00123 


20° C 


0.001155 


0.001175 


0.00119 


0.001205 


0.00122 


22° C 


0.00115 


0.001165 


0.00118 


0.001195 


0.00121 


24° C 


0.00114 


0.001155 


0.001175 


0.00119 


0.001205 



Density of Water. 



10° c. . . . 


0.99974 


15° C. . . . 


0.99915 


20° C. . . . 


0.99827 


11° c. . . 


0.99965 


16° C. . . . 


0.99900 


21° C. . . . 


0.99806 


12° C. . . . 


0.99955 


17° C. . . . 


0.99884 


22° C. . . . 


0.99785 


13° C . . 


0.99943 


18° C. . . . 


0.99866 


23° C. . . . 


0.99762 


14° C. . . 


0.99930 


19° C. . . . 


0.99847 


24° C. . . . 


0.99738 



28 SPECIFIC GRAVITY OF LIQUIDS. 



SPECIFIC GRAVITY OF LIQUIDS. 

This may be found by the specific gravity flask, Fig. 31, as follows : — 
Let b = weight in grms. in air of bottle empty. 

W= weight in grms. in air of bottle and distilled water at t° C. 
W x = weight in grms. in air of bottle and liquid at t x ° C. 
D t = known density of water at t° C. (See tables on p. 27.) 
A — weight 1 cc. of air at temperature, pressure and humidity of that in 
balance room at time when TFis found; A x = ditto for W x (p. 27). 
Reduction to Vacuo. — The weight of the flask need not be reduced, since 
W, W x , and b are very nearly equally affected, and thus the omission of the cor- 
rection would not affect the differences W— b and W x — b sensibly unless b were 
unusually large and A and A x considerably apart. 

The weight of water at t° filling flask at t° has a volume of about W—b cc, 
and therefore displaces (IF— b)A grms. of air, while the counterpoising weights 
displace A (W— b) -^8-5 grms. of air, 8-5 being the sp. gr. of the brass 
weights. Hence the corrected weight of this water at t° is 

w={W-b)U-^-\+(W-b)A. (1) 

And, similarly, since the volume of the liquid is the same (W— b cc.) as that 
of the water, its corrected weight is 

w x =(W x -b)(l-^A+(W-b)A x . (2) 



Reduction to Water at 4° C. — The weight of a volume of water at 4° C, 
equal to the volume of the bottle at t x ° (t and t x must be known to about 0.1°), 
may be found from w, as at pp. 26 and 27, as 

w[l + £($!-*)] -J- A- (3) 

The Sp. Gr. of the liquid at t x ° C. referred to water at 4° C. would therefore be 

-L>t 
This formula is the simplest which can serve where weighings are made to 
1 mgr. unless t x = t, or unless A and A x do not differ by more than 0.00001 grms., 
about, in which case the terms {W—b) A and (W— b) A x in (1) and (2) become 
constant. If a less degree of precision is desired, the expression may be cor- 
respondingly simplified. E.g., if the results are desired to 0.1 or 0.2 per cent, 
only, then the following reductions may be made : Substituting in (1) and (2) the 
value A = 0.00122 grms., which may now be assumed as constant, gives 

w= (TF-&)(l- 0,00122 > )+ ( W-b)X 0.00122= 1.0011 (W-b), and 

w x = 0.9999 ( W 1 -b) + 0.00122 (W- 6), whence 

Sp. Gr. = 0.9988 w \~ h Ih + 0.0012 (approx.). (5) 

1 W- b 1 + j8 fo - 1) K J W 

In this expression the weighings of W x and Tfare taken to 0.005 grms., t x and 
t are to be measured to 0.3° C, and £ may be assumed as = 0.000026. If a series 
of determinations of specific gravities is to be made, the process of computation 
may be still further simplified, since W—b and t will be constants. In results 
by either of the formulae (4) or (5) the sp. gr. determined is obviously for the 
observed temperature t x only. This temperature must, therefore, always be 
stated. Eeduction to standard temperature may be made as in (8), p. 27. 



mariotte's law (a). 



29 




Fig. 17. 



MARIOTTE'S LAW (a). 

(Pressures greater than 1 Atmosphere.) 

The apparatus consists of three glass tubes, A, B, C, of about equal dimen- 
sions, terminating at the top in glass cocks, and connected at D by a long rubber 
tube to a larger glass tube at E, which is open to the air, and slides 
upon a vertical rod carrying a scale, being clamped at any desired 
height by a set screw. The dried gases to be experimented upon 
are enclosed in A, B, C, respectively, and may be subjected to 
various pressures by raising or lowering E. The. direct object of 
the experiment is to determine the rate at which the volume of the 
gas diminishes as the pressure increases, or, in other words, the 
relation between the volume v and the pressure p when the tem- 
perature remains constant. Mariotte's Law asserts that the pressure 

is inversely as the volume, i.e., p=-; :.pv = c, where c is a constant. 

v 
This law, however, holds true only approximately in any case, and 
is more deviated from as the gas is nearer its point of condensation. 
In the apparatus to be used the tubes contain dry air, ammonia 
gas = NH 3 , nitrous oxide = N 2 0, the two latter being the more 
readily condensible. The tubes are labelled with the name of the enclosed gas. 

Suppose E to be in any position, as represented in the figure, and the 
mercury to stand at any corresponding height ki A, B, and C, e.g., at a in A. 
Then the pressure j> upon the enclosed gas (which, as there is equilibrium, must 
equal the expansive pressure of the gas upon the top of the mercury column at 
a, etc.) is measured by the mercury column equal to the difference of level 
between E and a plus the atmospheric pressure upon the top of the column at 
E; i.e., by the difference of level of E and a plus height of barometer at the 
time. The tubes are numbered upon the etched scale in cm. from above down- 
ward. If, therefore, from the reading of E be subtracted the difference between 
and o, the difference will be the vertical distance from o to E-, and if to this be 
added the reading a (== oa) , this sum will be the vertical distance aE desired, to 
which must be added the height H of the barometer. In order to avoid nega- 
tive quantities, it is better to perform these operations in the following order : 
OE -{- a— Oo + H=p. As the zero points on the tubes could not all well be 
adjusted to the same height, the value of Oo for each has been determined by 
levelling, and is as follows : A, Oo = 85.24 cm. ; B, Oo = 87.05 cm. ; C, Oo = 82.30 
cm. Compare these figures with those given on the apparatus. The value of p, 
as well as of v, has, of course, to be calculated for each gas at each position 
of E. 

To find the volume v. If each tube were of uniform cross-section through- 
out, and the zero point of the scale were rightly placed, then the scale reading 
would give the volumes directly in arbitrary units whose value would be the 
capacity of one unit length of the tube. Neither of these conditions can be 
fulfilled, and it therefore becomes necessary that the tube should be calibrated 
beforehand, i.e., there must be found beforehand by measurement the volumes 
up to different marks upon the scale. This has been clone for each of the tubes, 
and the results are shown in the curve accompanying the apparatus. To find 
the volume corresponding to any reading a, look out in the horizontal axis the 
number a, and find the ordinate of the proper curve at that point. The sum of 
this ordinate and the scale reading a gives the volume v in an arbitrary unit 
similar to that just described. 



30 mariotte's law (6). 

The experiment consists in placing E first as low as is consistent witli having 
the mercury in A, B, and C come upon the graduation, and in reading all four 
columns ; then raising E about 50 cm. , reading again, and so on, taking five or 
six different positions of E. The barometer should be read before, during, and 
after the experiment, to obtain proper values of H. The meniscus at E is to be 
read by raising and lowering the eye in front of the tube until it appears to be 
on a level with the horizontal upper surface of the mercury, and taking the scale 
reading to tenths of a mm. The meniscus in each tube is best read by bringing 
the black-paper clasp down nearly to the top, but not covering the division 
next above it, and then bringing the eye up nearly to the plane of the lower edge 
of the paper, and estimating the tenths of a division (mm.) by the eye. 

It is essential that the temperature should remain very nearly constant 
during the whole experiment ; hence any handling of the tubes is to be avoided. 
No temperature corrections are to be applied, as for present purposes they 
would be unnecessary, unless a considerable change of temperature occurs 
during the observations. 

If Mariotte's Law held true, then, as before stated, pv=c. To show how 
the gases vary from this law, calculate for each observation upon each gas the 
value of pv. Plot points with values of p as abscissas, and of pv as orclinates. 
The resulting lines will be found in each case very nearly straight, but with 
differing inclinations. The line would be straight and parallel to axis OX if the 
gas followed Mariotte's Law, and the greater the deviation the greater the 
inclination of the line. Find, therefore, the tangent of the inclination of the 
lines to OX. A negative value of this tangent indicates, of course, that the gas 
is more compressible than the law indicates. As the volumes of the gases are 
not numerically equal, the values of the tangents will not be strictly com- 
parable. If, however, the tangent for each of the more compressible gases be 
multiplied by the ratio of the pv for air to pv for that gas, these products 
will be directly comparable. 

As systematic work is of material assistance in reducing the labor of compu- 
tation, and avoiding numerical and other mistakes, the student should use 
especial care in the arrangement of these computations. 



MARIOTTE'S LAW (b). 

(Pressure less than 1 Atmosphere.) 

A graduated tube il, closed at the top filled with dry air, slides with slight 
friction in a guide de. The enclosed air, in virtue of its expansive force, exerts 
a pressure upon all parts of the enclosing vessel. Hence, at the top 
a of the mercury column, in order to bring about the equilibrium 
which there exists, there must be an upward pressure exerted through 
the mercurial column equal in amount to the downward pressure of 
the gas upon the column. This upward pressure comes from the 
pressure of the atmosphere upon the mercury in the cistern at c, and 
its amount is measured by the height of the barometer at the time of 
the experiment minus the column ac, since the latter opposes the 
atmospheric pressure. The barometer column and ac must, of course, 
be measured in, or reduced to, the same unit. There is thus an en- 
closed mass of air in ai, whose volume and pressure can be measured 
and varied by sliding de. 
Fig. 18. Raise de until i touches the stop above the top of the tube. Eead 

the position of the top of the meniscus at a to tenths of the smallest division 



«7d 

6 



BTJKSEN PHOTOMETER. 31 

of the graduation. To do this, lower the paper clasp, which will be found on the 
tube, until it is nearly tangent to the top of the meniscus, thus cutting off the 
troublesome reflections from the meniscus, and then, looking as nearly as pos- 
sible perpendicularly to the tube, estimate the fraction of the division by the 
eye. Call this reading a. 

To find the height ac, set the rod with its lower point just in contact with the 
mercury at c, and read the position of the upper end b upon the scale to tenths 
of a division. This reading may be best made by moving the eye until the image 
of the rod formed by reflection from the mercury surface is just in the line of 
sight across the end b of the rod itself, and then reading the position of this 
image, calling this reading b. A white paper may be held so that its reflection 
will make a good background for the image of the rod. Then withdraw the rod, 
place it alongside the tube, and read its length be to one-tenth of a scale division. 
Then, as the scale is numbered downwards from i, ac=bc+ (b — a). 

Subtract the distance ac from the height of the barometer taken at the time 
(expressed in the same unit) , and the difference will be the pressure p of the gas. 
Next, to find its corresponding volume v. This is given approximately by the 
scale reading of a; but this requires a correction for irregularities of the tube, 
which is given by the "curve of calibration" accompanying the instrument. To 
find the correction, look out on the curve the abscissa corresponding to the 
scale reading, and find the ordinate of the curve for that abscissa. Acid this 
ordinate to the scale reading, and the sum will be the volume v of the gas, 
expressed in an arbitrary unit about equal to the average volume contained 
between two successive cm. marks on the tube. 

Determine the values of p and v for six or eight different positions of the 
tubes, changing the positions as much as the apparatus will allow. 

By Mariotte's Law, p varies inversely as v ; .-. pv = c, where c is a constant. 
Hence, if a curve be plotted with values of p as ordinates, and of v as abscissas, 
it should appear as part of a rectangular hyperbola. If a curve be plotted with 
values of p as ordinates and reciprocals of v as abscissas the line should be 
straight, and the straightness will serve to test the accuracy of the results. 

Draw both curves on the same sheet. 



BUNSEN PHOTOMETER. 

These notes replace the description on pp. 135-138, "Physical Manipulation," 
but the paragraphs there given under "Apparatus" furnish some further details 
in regard to various parts and accessories of the photometer which are not used 
in this experiment. 

The Bunsen photometer is a convenient device for the measurement of the 
ratio of two lights of the same, or nearly the same, color. A disk of white 
paper is covered with grease except over a central circle of about an inch diam- 
eter. This disk is placed in a line joining the two lights whose ratio is to be 
measured, and is perpendicular to that line. Viewed from one side, the central 
spot of the disk will appear brighter than the surrounding portions when the 
illumination of that side is greater than of the reverse, darker than the surround- 
ing portions wheu the illumination is less than of the reverse. Thus, by moving 
the disk forward and back between the two lights, it will be found that at some 
point the spot and surrounding portions will appear of the same brightness, and 
will be indistinguishable from each other, or nearly so, the total disappearance 
of the light depending upon the complete equality in color of the two lights, aud on 
the nature of the disk, as well as upon the equal illumination. When this disappear- 



32 BUNSEN PHOTOMETER. 

ance of the spot is attained, it is known that the disk must be equally illuminated 
on either side by its respective light. Then, since the intensity of illumination 
at any distance from a given source of light is inversely as the square of the 
distance from that source, the ratios of the two lights must be directly as the 
squares of their respective distances from the disk. In the photometer, as ordi- 
narily used for gas measurements, the lights are placed at opposite ends of a bar 
100 inches in length, upon which the disk slides by means of car or carrier of 
some convenient form. If a scale of inches be placed along this bar with the 
zero at one light, and if a be the observed reading of the disk, then the ratio of 
the light L at the further end to that C at the zero end of the bar will be 

£_(100_a)2 



Instead, however, of a scale of inches, we may have one graduated to give the 

above ratio directly. To make this graduation, we have but to put for — success- 
es 
ively 1, 2, 3, 4, etc., and any intermediate values desired, and solve for a, marking 
the point thus found upon the bar. In the apparatus used for this experiment 
both scales will be found, but at least one result should be computed from read- 
ings on the scale of inches. In most measurements a " standard " sperm candle 
is placed at the zero end of the scale, and the light of this caudle is taken as 

unity, so that the ratio becomes — , i.e., gives the " candle power" of the light 

measured, the necessary corrections being applied as given below. The appara- 
tus must, of course, be located in a completely darkened room, the walls of 
which should be dull black to avoid reflections. Any auxiliary lights used for 
readiug must be small and carefully screened from the disk. 

In the present experiment, the candle power of illuminating gas burning at 
various rates in an argand burner, is to be measured in terms of the corrected 
standard candle, with a view to obtaining both the nominal caudle power of the 
gas under standard conditions, and the relative efficiency of the burner used 
when consuming gas at various rates. 

Candle. — The standard candle is a superior quality of sperm candle, six to 
the pound, and so proportioned as to burn nominally 120 grains of wax to the 
hour. The light from such a candle, when burning at this rate in open air, is 
tolerably uniform and reproducible, and is assumed as the unit of light in most 
photometric measurements. It is further assumed, for correcting for unavoid- 
able irregularity in the rate of consumption, that the light given out by the 
candle is proportional to the amount of wax burned per minute. This assump- 
tion is, however, not strictly true ; and these variations in efficiency, or illumi- 
nating power of the candle, make it an unsatisfactory standard, though perhaps 
the best yet generally available for the purpose. Owing to this inequality in 
ratio, it is necessary to weigh the candle, unless some means is at hand for using 
the average light, or getting an average result by the use of a large number of 
candles. The form of balance often used for this purpose carries the candle on 
a short arm of the beam, and has for the other arm a longer rod notched in 
grains. The end of this arm carries a pointer or index, which moves over a 
vertical scale. This form of balance, though sensitive to perhaps the tenth of a 
grain, is not capable of weighing thus closely on account of the small distance 
between the notches ; and the use of the scale at the index end, intended to 
also indicate grains, is at best unsatisfactory. The precision of the weighings 
are greatest when made by noting the exact time interval of burning a whole 
number of grains, as described further on. 



BTTNSEN PHOTOMETER. 33 

Gas. — The State law of Massachusetts requires, at the present time, that 
when burning at the rate of 5 cu. ft. per hour under favorable conditions, as in 
an argand burner with proper air supply, illuminating gas should give out a light 
equivalent to that of at least 15 candles. The light given out by gas under 
such conditions, and called its " candle power," is nearly, but not exactly, pro- 
portional to the rate of consumption when this rate does not vary widely from 
5 cu. ft. per hour. This proportion, however, does not hold when the rate is much 
smaller or greater than the normal for the given burner, owing to the variation in 
the efficiency of the burner. The rate of consumption of gas while the pressure 
remains constant will be found quite uniform. It is easily measured with 
precision by the " wet meter" provided with this experiment. The water in the 
meter should be kept up to the reference mark on the glass gauge. On the dial 
the long hand makes one revolution for every tenth of a cu. ft. of gas pass- 
ing through the meter. The outer circle over which this hand travels is 
divided into 100 parts, and thus reads to 0.001 cu. ft. The small hands count up 
these revolutions, so that one division on the circle marked "one foot" indicates 
0.1 cu. ft., one revolution being one cubic foot; and so on for the other circles. 
The gas jet in front of this meter should be turned very low. 

Time in the experiment may be taken from the clock which rings warning at 
7 seconds before each minute, and a gong stroke just on the minute, for con- 
venience in the observation of the meter. 

Disk. — It may be found at first somewhat troublesome to set this with pre- 
cision. Repeated trials with it should be made before beginning the experiment 
in order to acquire facility in its use. It will be found that the disk appears 
slightly different according to the position from which it is viewed, and should 
be looked at from a definite direction, e.g., at an angle of about 45° with its 
surface. Also, it will be found that at first different results are obtained in 
setting by pushing the disk away and by drawing it towards the observer; and 
different results on opposite sides of the disk. After some practice these 
differences largely disappear; but precision is increased by setting in both, 
directions and from both sides, and taking the mean of the four readings thus 
obtained. During a measurement, disk observation should be multiplied as much; 
as possible to aid in the elimination of errors of observation. 

In making a measurement there are thus to be observed the time-rate of" 
consumption of the gas and of the candle, and the position of the disk. And as. 
the rates are both liable to change, it is desirable to make the observations on: 
them both and on the ratio to extend over the same time interval. This is- 
readily accomplished as follows : Light the candle and the gas, and allow the- 
candle to come to its normal condition of burning. Then place the balance rider 
in a notch near the zero end of its scale, and place small shot in the scale pan 
beneath the candle until the index of the balance beam goes up nearly to the top 
of its scale. Then wait until the index (which must be kept in vibration) , settling 
as the candle burns, swings equal distances on each side of the middle (zero) 
division. Note by the clock the exact time in minutes and seconds. Record 
this and the reading of the rider on the balance arm. Then turn to the meter 
and disk readings, allowing the candle to burn without attention during the 
ensuing five minutes. At each successive stroke of the gong, for the three or 
four minutes next succeeding the first balancing of the candle, read the gas 
meter, and in the interval set and read as frequently as possible the disk. As 
the end of the five minutes approaches return to the balance, move the rider up 
several notches (10 or more, probably), enough to again allow the index to 
vibrate about a point above the middle point of the scale, note the new position 
of the rider, wait until, as at first, the index swings equally about the middle 



34 ABSORPTION SPECTRA. 

scale division, and at that instant again note and record the time. The differ- 
ence in times thus obtained gives the interval during which the candle has 
burned the number of grains found by subtracting the two corresponding rider 
readings. From these data the rate of consumption of the candle in grains per 
hour can be computed. Call this c. From the meter readings may be computed 
the rate of burning of the gas. Call this g. Take the mean of the computed 
ratio obtained from the disk settings, and call this r. Then the value which would 
have been found for this ratio, which would then be the candle power of the 
given flame, had the candle been burning at the normal rate, would have been 
(under the assumption above stated) 

120 
Now if the gas consumption had been at the rate of 5 cu. ft. per hour, assuming 
the burner to have been a proper one, this L would be the desired " caudle 
power of the gas." If the rate is other than 5 ft., but not widely different, the 
c. p. of the gas will be approximately given by the expression 

L = r. -°-.* 

120 g 

and the values of L thus found corresponding to various rates g will afford a 
measure of the relative efficiency of this burner, and will serve equally for com- 
parison with the same quantity deduced from other burners. As this is some- 
what arbitrary, however, in the assumption of the 5 ft. rate as standard, a 
better way is to use as the measure of the efficiency of the burner the light given 

c 1 
out per cu. ft. of gas burned, i.e., E = r • — • -. 

120 g 

Measure r, c, and g for the given burner for at least four rates of burning of 
the gas, ranging from the lowest to the highest flame obtainable without smok- 
ing. Compute from each the value of E. Plot points, with values of E as 
ordinate, and of g as abscissas. If the efficiency were uniform, the line through 
them would be parallel to OX. Its curvature and inclination show the relative 
efficiency at different rates. Find the ordinate of this curve when g = 5, and com- 
pute from this the value of L corresponding to these measurements. Students 
unfamiliar with the graphical method may compute L by the last expression 
above given for it. 



ABSORPTION SPECTRA. 

The light entering AB in the direction of the arrow passes through the 
lower half of the vertical slit 8 through the lens D to the prisms H, 1, J, K, L, 

of which H, J, and L are of crown glass, 
and K and I of flint glass. By this combi- 
nation the rays of various colors undergo dis- 
persion, but emerge from L parallel to their 
original direction. Thus, on looking in at 
E, the spectrum of the light coming in at S 
is seen. A small total reflection prism C covers the upper half of the slit S, 
and reflects upon it light coming in as shown by the arrow between Q- and B. 
Thus at .Sis seen another spectrum due to this light, and situated above the 
first. In the smaller side tube at M is placed a scale photographed on glass. 
The light coming from this scale passes through the lens N to the reflecting 
prism P, whence it is reflected to the surface of the prism L, and thence through 
E appearing below the two spectra, and forming a scale of reference. The slit 




INDEX OF REFRACTION. 35 

& is adjustable in width by turning the revolving rim G. Starting with the slit 
very narrow, first focus the lens D by drawing the inner tube back and forth 
until the Fraunhofer dark lines are clearly visible, pointing the spectroscope 
toward a window. Direct sunlight is not necessary, and is usually too bright. 
Then slide O back and forth until M is clearly focused. The instrument is then 
ready for work. 

Locate on the scale as many of the Fraunhofer lines as can be readily seen, 
recording their positions and their letters as found from the map accompanying 
the apparatus. A brief description of these lines may be found at p. 53, " Physical 
Manipulation," vol. i., and in the Physics Lecture Notes. Notice that these lines 
are most clearly seen when the slit is very narrow, and that when sunlight comes 
in through the front and side apertures at the same time, the various lines oc- 
cupy exactly the same positions in the two spectra. 

The colors of some bodies are produced by the absorption of certain portions 
of the light falling upon them, and this absorption may be conveniently studied 
with this instrument. If a mirror or piece of white paper be placed so as to 
throw light in the direction of the arrow at G through the prism C, the upper 
spectrum will be formed, and will be constantly in the field for reference. If a 
piece of colored glass be held in front of AB, between the window and the slit, 
it will absorb part of the colors of the sunlight, transmitting the remainder. 
The transmitted light will be dispersed by the prism, and its spectrum will 
appear below the reference spectrum as seen through E. In this way the trans- 
mitted light may be analyzed. Examine all the specimens of glass and the 
colors of the transmitted light in each case. The absorption spectra of liquids 
may be similarly studied. 

Record the appearance of the spectra of light passed through the solutions of 
clidymium sulphate, chlorophyll, rosin, etc., contained in the small bottles. It is 
sometimes more convenient to note the dark bands instead of the light trans- 
mitted. A wedge-shaped glass bottle containing a solution of potassium perman- 
ganate shows well the effect of increasing the thickness of the absorbing layer. 

Finally, examine the spectra of the gelatine films between the square glass 
plates. These films are colored by various aniline clyes, the names of which are 
given on the card accompanying the plate. The best way of recording the 
results in this experiment is by drawing sketches of the spectra, one above 
another, with a scale beneath, and with the dark bands located as observed upon 
the scale of the instrument, placing the name of the substance opposite its 
spectrum. 



INDEX OP REFRACTION. 

The index of refraction of a glass prism may be found by the use of the 
optical circle described in "Physical Manipulation," page 141. In all measure- 
ments both verniers must be read, and the correction applied for eccentricity, as 
in the experiment on Eccentricity of Circles in these notes. 

Also, if the prism be made as in the sketch, with a 
cavity between two of its faces, the ends of cavity being 
closed by suitable pieces of plate glass, the index of re- 
fraction of liquids may be obtained, observations being 
made as in the following directions for the glass prism. 

The light from a sodium flame in front of the collima- FlG * 20 * 

tor AB passes through the slit, and emerges as a parallel beam from the lens 
at B, the slit having been previously adjusted to be in the principal focus of this 





36 LAW OF KEFRACTION. 

lens, as described, page 143, " Physical Manipulation." This parallel beam meets 
the prism, which is placed in the position shown, and emerges, after refraction, 
in the direction of the dotted line. This emergent beam is parallel if the faces 
of the prism are plane, as they should be, and is brought to the eye by the 
observing telescope, and thus may be seen, either with or without the latter, as 
an image of the slit of the color of the sodium flame. By 
turning the prism alternately in one and in the opposite direc- 
tion about its vertical axis, and observing the refracted image 
of the slit, either directly or through the telescope, it will be 
found that the emergent ray will move from various positions, 
Fig. 21. c'd' up to a limiting one cd, never approaching nearer than this 

to abe. The ray then undergoes, as is seen from the figure, 
the minimum deviation, and the angle of incidence of the ray ab is then exactly 
equal to the angle of emergence of the ray cd. For this position denoting by 
a the refracting angle of the prism, and by D the angle of deviation, i.e., the 
angle between the line abe and cd, it may be shown that the index of refrac- 
tion n is 

sin h (D + a) 
n = — - 

sin- 
2 

To measure D, set the prism at the minimum deviation ; adjust CD so that the 
cross-hairs bisect the image, and read both verniers. Then reverse the prism so 
that the emergent ray shall have the direction c'd'. Fig. 22. Bisect the. image 
as before, and read both verniers. The corrected difference will be 
^^T/^* twice the angle D. Instead of the second observation, the prism 
^-^^^—- i s sometimes removed after the first, and the observing telescope 
^sS\^| brought into the line nb, the direct image bisected, and the cor- 
rected vernier reading taken as a zero-reading to be subtracted 

Fig 22. & 

from the first reading, giving D. The former method is, however, 
somewhat more accurate. 

The value of n thus obtained is the index of refraction of the substance for 
the sodium line, i.e., light of that wave-length. If the index for light of any 
other color (or wave-length) is desired, it is best obtained by finding the devia- 
tion of some suitable dark line (Fraunhofer line) in the solar spectrum, or in 
some cases by the use of some other than a sodium salt in the flame. 

Give in the note-book a demonstration of the above formula for n. 



LAW OP REFRACTION. 

A slip of glass with parallel faces is mounted upon a revolving table 
carrying a divided circle read at a fixed index. A millimetre scale d and eye- 
hole e are so placed that a line drawn from e through the zero 
~+2 of the scale would pass through the centre of revolution of the 

table. On looking through e, half the scale d is seen above and 
half through the plate. As the table is revolved in a right-handed 
direction, the lower half of d will appear to move to the left, 
so that the lower part of the divisions at the right of the middle 
may thus be successively brought to coincide with the upper half 
Fig. 23. of the middle line. When this occurs for any division, the ray of 
light, coming from this to the eye, is shifted parallel to itself from 
its original position by an amount equal to the distance D between that division 




THERMOMETRY. 



37 



and the middle one. Calling T the thickness of the glass (in this case 8 mm.), 
i the angle of incidence, and r the angle of refraction, then 

D 



For, in Fig. 24, 



tan r — tan % 
T 



AB = 



I 1 cos i 
D 



cos r sin (i — r) 



t\ sin ( i V s ) • • 

whence — == - — ^ L ; but sin (i — r) = sin i cos r — cos i sm r, 



cosr 



D 



whence yp = sin i — cos i tan r ; or, tan r 



cost 



= tan i — 



Tcosi 



Take readings of the circle when the lower half of the first division on each 
side of the zero is brought to coincide with the upper half of the zero mark, 
the object of using both marks being to eliminate the zero 
error of the circle. Then turn the circle and plate through 
180°, and take similar readings to eliminate eccentricity. 
Take the mean of this set of four readings as the value 
of i corresponding to this value of D. Make similar ob- 
servations for each of the first five millimetre marks on 
each side of the middle line. Compute from the above 
formula the value of r corresponding to each of the five 
mean values of i thus found. Construct a curve, using as 
abscissas the sines of these values of r ; and as ordinates, 
the sines of the corresponding values of i. This should 
be a straight line, and will make with the axis of abscissas an angle whose tangent 

^°-^ = n the index of refraction of the glass, thus proving the law that -^52 

sinr 




Fig, 24, 



sinr 

= a constant. 



Determine n from the curve drawn. 



THERMOMETRY. 

Each student taking this subject will, — 

1st. Read carefully all the notes on Thermometry. 

2d. Take a suitable thermometer, record its number in the note-book, and 
attach to the case a tag bearing his name, so that the same thermometer 
may be used in all subsequent work so far as desirable. 

3d. Find its calibration errors, using a thread about 3 cm. in length. 

4th. Arrange for the thermometer a suitable bath for determining the 
stem exposure correction, to be used in subsequent work; e.g., "Expansion 
of Liquids," etc. 

5th. Find the reading in ice, and then in steam, as described in the notes, 
and compute the value of a. 

6th. In some cases (ask the instructor), find the deviation from the air 
thermometer. 

The form of mercurial thermometer capable of the greatest accuracy in use 
is that having upon its stem an engraved scale of equidistant lines, e.g., milli- 
metres. If such an arbitrary scale is used, the corrections and reductions are 
in general exactly the same as for the scale of degrees, except in the stem 
exposure correction, where the slight difference is as shown in the notes. 

The observed reading R of any mercurial thermometer is subject to several 
sources of error, for which the proper corrections must be applied before the 



38 THERMOMETRY. 

temperature of the bulb at the time of observation will be correctly known. 
The chief of these — to be applied in the order named — are stated in brief on 
this page, and in fall in the following notes. The formulae given on this page 
are for thermometers having scales of degrees. The formulas for arbitrary- 
scales will be found at the proper places in the following pages. 

1st. Calibration. — For each reading of the thermometer, take from the 
curve of calibration errors (or corrections) the error (or correction) corre- 
sponding to that reading. Subtract this error from (add the correction to) 
the reading : 

B' = R-e or B + c. 

2d. Temperature of Exposed Stem. — Let s° = temp, of stem, found by auxil- 
iary thermometer; n — number of degrees length of mercury thread in stem at 
s°; and 0.000156 = coefficient of apparent expansion of mercury in glass. Then 

R" = B' + 0.000156 (22' - s°) n. 

3d. Reading in Ice, and Value of One Unit of Scale Division. 

B ' = observed reading in ice, corrected for calibration. 
RJ = observed reading in steam, corrected for calibration. 
T 8 = computed temperature of steam. See p. 42. 
The value of one unit of scale in degrees centigrade is 

**§=%> ••• *"'=«(*"-*.'). 

4th. For Difference between the mercurial and air thermometer, and, when 
the limiting precision is desired, for difference between the air thermometer and 
the absolute scale of temperature. This total correction will be a function of 
B'" for each particular thermometer, and may be represented by <p(R"'). 

T=B"' + $(&"). 

1st. Calibration. 

No thermometer tube of perfectly uniform bore can be -obtained ; hence it is 
essential that the scale etched or engraved upon the glass stem (a scale engraved 
or etched upon the tube is the only one suitable for exact physical or chemical 
work) should be so spaced as to mark off equal volumes of the capillary, or 
that it should consist of equal linear spaces, so that the points corresponding 
to equal volumes may be determined by calibration. Thermometers with approxi- 
mately accurate scales of the first kind constitute the main supply in the market, 
for makers are aware that their instruments are ordinarily judged by their appar- 
ent and not by their corrected indications. Very few such thermometers are to be 
relied upon as possessing a maximum calibration error of less than one-half of 
the smallest division of the scale, and will frequently be in error from this cause 
by more than one whole division. Moreover, such thermometers can not be 
calibrated by any ordinary method to arrive at a precise determination of errors, 
for the divisions upon the scale consist of a series of groups of equal linear spaces, 
the spacing in different groups, however, being unequal. This results from the 
method of graduating, the stops of the dividing engine or other tool being 
changed only at intervals, so as to make the average error not to exceed some 
specified amount. In the cheaper grades of thermometers this work is very 
roughly clone. Any thermometer with a scale of equal linear parts, whether of 
approximate degrees or of arbitrary lengths, may be readily calibrated by the 
following method so that no error exceeding 0. 1 of a division shall exist uncor- 
rected at any place. As calibration is ordinarily to be performed upon com- 
pleted thermometers, such only will be now considered. 



THERMOMETRY. 39 

The principle of all methods of calibration consists in noting the exact 
length of a thread of mercury of suitable length placed successively in different 
parts of the capillary, and from this determining the points separating equal 
volumes. 

To separate a thread of the desired length, the method described in the 
" Physical Manipulation," p. 76, may be used, or the tube may be placed in a 
horizontal position and tapped somewhat sharply upon the upper end with a 
piece of soft wood. This will cause the thread to break into pieces, from 
which may be made up the proper length. In order that the calibration may be 
carried to the lower end of the scale, it is usually necessary to remove some of 
the mercury from the bulb into the small bulb at the top of the instrument. As 
this operation is most readily understood by witnessing it, no description will 
be here given, but each person will be shown individually. 

Method of Calibration. 

The calibration correction is usually to be found for every 2 cm. to 3 cm. of 
the tube, the latter being sufficiently close for most work. Separate a thread 
of mercury about 3 cm. long, the actual length within a few millimetres being 
of no consequence. If the mercury should project from the bulb into the 
stem more than this amount, it will be necessary first to separate some of the 
mercury from the column, and transfer it to the auxiliary bulb at the top, so 
that it may not interfere with subsequent operations. 

Set the thread with its lower end at or near the beginning of the graduation. 
It is often sufficient to start at the zero-point of graduation, however, when the 
scale is not to be used below that point. It is frequently best to carry the 
calibration thread at once to the top of the capillary, and to make the observa- 
tions as the thread is moved from the top to the bottom of the tube, instead of 
the reverse, thus avoiding the possible inconvenience of having the mercury from 
the bulb accidentally enter the capillary and unite with the thread. The descrip- 
tion supposes the readings to be begun at the lower end of the tube. Call the 
reading of the lower end of the thread l x ; that of the upper end u v Move the 
thread less than 1 division and read again, finding thus l 2 and u 2 . Move the 
thread half its length or less and read ? 3 and u 3 . Move less than 1 div. and read 
Z 4 and u v So continue throughout the whole length of the graduation, increas- 
ing the number of settings, and repeating the whole series in reverse order if 
higher precision is desired. Avoid, as far as convenient, taking readings with 
an end of the thread apparently just at a line of the scale, as the width of a 
line, even in the best scales, is a source of considerable error. 

It is essential that the temperature of the capillary should be as uniform 
as possible throughout these measurements, and the tube must therefore be 
handled as little as possible during the work, and then only by its ends. In 
very careful work further precautions are necessary. 

Now u x — l v w 2 — l 2 , etc., will give a series of lengths of the calibrating thread 
in all parts of the tube. Before reuniting the mercury thread with the main 
portion, plot points with abscissas l lf 7 2 , etc., and ordinates u-^ — l^ u 2 — l 2 , etc., 
the corresponding lengths of thread, and draw a smooth curve through the 
points thus obtained. This line will give a general idea of the form of the 
capillary bore, and should any part of it show considerable irregularities, the cor- 
responding portions of the tube should be at once reexplored with the thread. 

Select arbitrarily as the starting point A for the computation of errors any 
point near the lower end of the tube, but not so near the bulb as to be in the 
expanded end of the bore. Find by interpolation on the curve the ordinate u' 
corresponding to the abscissa A ; then, with the abscissa A + u', find the corre- 



40 THERMOMETRY. 

sponding ordinate u" ; with the abscissa A + u' + u" the ordinate u'", thus con- 
tinuing to the upper limit of the graduation. These points, A, A-]- u', A + u' + u", 
etc., upon the graduation are separated by equal volumes of the capillary, and 
if the tube were of perfectly uniform bore, the intervals u x — l x , u 2 — Z 2 , etc., would 
be all numerically equal. Suppose, for convenience of explanation, that A = 0, 
and that «j = 10.3, then u 2 , if the tube were uniform in bore, should, be 2 x 10.3 = 
20.6. But suppose by the reading it is found to be 20.8, then the error at that 
point clue to irregularity of the bore, would be 20.8 — 20.6 = +0.2, and so on. 
But by this process of computation the numerical values of the. errors may in 
some cases become inconveniently large, and this difficulty may be. avoided by 
using as the calibration unit instead of u x — l x the average length of thread 
over a considerable length of tube. For this purpose select any one of the 
upper points of equal volume obtained as above (viz., A + u', A + u' + u", etc.), 

and call this point B. Then A + u' + u" + \-u nth = B. There are thus n 

spaces of volume equal to that of the calibrating thread in this interval B — A, 

B — A 

and the average length of the thread in this part of the tube is . Hence, 

n 
the numerical error at A and at B will be zero, in the same way as at A and 
A + u' in the previous illustration. Thus, the true reading at the point 

A is Ay 

A+u' « A + -(B-A), 

n 

A + u'+u" " A + -(B-A), 

n 



B " B. 

These readings, if subtracted from the interpolated readings A, A + u', A+u'+u", 
etc., give the errors at these latter points, and these are to be computed. Thus, 
the error at 

A is 0, 

A + u' " A + u'-[a + ±(B-A)] = u'-±(B-A), 

I n ) n 

A + u'+u" " u'+u"--(B-A), 

n 



B " 0. 

In selecting the points A and B, it is often convenient to take for the former 
the zero-point of the graduation, and for the latter a point near the top of the 
graduation, so that n shall be as large as convenient. The result will be of 
equal precision in any case, but the correction will ordinarily be numerically 
greater when n is small than when it is large. The error or corrections 
(= — errors) are, for purposes of interpolation, most conveniently represented 
graphically by a smooth curve drawn through points plotted with abscissas 
proportional to the direct readings, A, A + u', A + u' + u", etc., and ordinates 
proportional to the corresponding corrections. 

Should it be desirable to increase the accuracy by a second calibration, with 
a thread of different length, it is only necessary to take one of approximately 
an integral part of (B — A), and when the final curve of errors is drawn, make 
the error at B equal to zero, distributing the difference between the two curves 
at B among the errors at intermediate points proportionally to their respective 
scale readings. In other words, shift the axis of the second curve of error so 
that it shall make the error at B zero. 



THERMOMETRY. 41 

2d. Temperature of Exposed Stem. 

In temperature measurements it is seldom possible to have the stem im- 
mersed in the bath of which the temperature is to be found. Thus, a portion of 
the mercury and stem are at a temperature different from that of the bulb. The 
reading of the thermometer will, therefore, require correction by an amount 
proportional to the apparent expansion in glass of this exposed portion of the 
mercurial column when heated (or cooled, as the case may be) from its actual 
temperature to that of the bulb. 

I. For Thermometers with Scale in Degrees. — Let 

R' = observed reading R corrected for calibration. (Approx. degrees.) 
s° = observed temperature of exposed part of stem (In approx. degrees) 

by an auxiliary thermometer. 
n = number of degrees of stem at temperature s°. 
T = true temperature of the bulb (Desired) . 
0.000156 = coefficient of apparent expansion of mercury in glass. 
Then the reading corrected for stem-exposure will be 
R"= R'+ 0.000156 (T— s) n ; 
or, since T and R' are nearly equal and T is unknown, this is found approx. as 
R"= R'+ 0.000156 {R'~ s) n. 

If the last term of the second member of this equation becomes large, it may be 
necessary to treat this value of R" as a first approximation only, and, substitut- 
ing it in place of R' in the parenthesis, to compute a second value of R" . It 
frequently occurs that R'—s is as large as 60° when R'= 100° C, and n = 100°. 
In such a case 0.000156 (60) 100 = 0.94, nearly one degree. At higher tempera- 
tures the correction often becomes ten times this amount. In the determination 
of n the calibration correction may usually be neglected. 

To obtain s the best way is to surround the stem of the thermometer with a 
thin glass tube about an inch in diameter, closed at the bottom with a thin 
section of a rubber stopper. Read the temperature of the liquid in this 
tube by an auxiliary thermometer placed successively at different heights 
in the tube. A vertically moving stirrer will generally serve to keep the 
temperature of the liquid sufficiently uniform, but when this does not 
suffice, more than one auxiliary thermometer must be employed. For the 
liquid, water may be used when T does not exceed 100° C. ; above that 
glycerine or some non-volatile transparent mineral oil. Rubber will 
withstand the action of a hot oil bath for some hours at 250°, but is not 
serviceable at such high temperatures. 

A rough approximate calculation, made beforehand for any case, will FlG - 25 - 
show whether this correction need be applied. It should be used with the high 
temperature thermometer in all subsequent calorimetric work. 

II. For Thermometers with Arbitrary Scale. — See page 43. 

3d. Reading in Ice, and Value of One Unit of Scale Division. 

I. When the thermometer is capable of reading in ice and also in steam, or, 
as ordinarily expressed, can have the " Freezing and Boiling Points " determined, 
the value of the unit of scale division may be determined by reading in ice and 
in steam at known pressure in suitable succession as described below. This 
method applies equally to a scale of degrees or of arbitrary division of equal 
length, remembering that the calibration corrections must be applied to every 
reading. The reading in ice should usually precede that in steam. 



42 



THERMOMETRY. 




To determine the " freezing point," or 0° of the centigrade-scale = 32° of the 
Fahrenheit scale, place the thermometer in a considerable mass of finely pounded 
ice, with the zero-point (a centigrade thermometer always supposed) just visible 
above the ice. Then pile up fine ice around the stem for a distance of an inch 
or more above the zero, so that the stem may be thoroughly cooled. Allow the 
instrument to stand for about ten minutes, then remove the ice in .front of the 
zero-point, and read as accurately as possible the position of the mercury, taking 
care always to look perpendicularly to the tube. Replace the ice, and, in a few 
moments, repeat the observation, continuing such readings till they appear 
constant. Let this reading be called R Q . Next determine the reading of the 
thermometer in steam, reading at the time the barometer 
and attached thermometer. The apparatus to be used for the 
readings in ice and steam are those shown in Fig. 26, and 
described in most text-books on Physics. (" Physical Manipu- 
lation," ii., 73 and 74.) Almost any vessel about 4 inches in 
diam. and 6 inches or more deep will answer for the ice, 
though it is better to have the water drain away. 

A convenient and safe arrangement for holding the ther- 
mometer is shown in Fig. 26. The stem fits tightly into a 
a thin section of rubber stopper, but runs loosely through 
a stopper, which is fitted permanently into the top of the 
heater. The thermometer bulb should be about two inches above the surface of 
the water, and within the double jacket. The water should be boiling, but not vio- 
lently. The top of the mercury column must be below the stopper, as shown, 
so that the whole of the mercury column is in the steam, except when readings 
are to be made, for which purpose the thermometer may be raised until the mer- 
cury column is just visible above the stopper. 

Let BJ = observed reading in ice corrected for calibration. 
BJ = " " " steam " " « 

T s = computed temp, of steam as found from the reduced barometer 
reading either by tables (Smithsonian Coll. of Meteorological Tables ; Landolt's 
Tables; Wiillner's Lehrbuch der Exp. Phys., iii., etc.) or by the approximate 
formula T s — 100+ g% (H— 760) where H= corrected barometric height in mm. 

The degree centigrade by the mercurial thermometer may be defined as that 
temperature interval which would produce in the mercury of the given ther- 
mometer 0.01 of the apparent expansion observed when this thermometer is 
heated from 0° C. to 100° C. (as elsewhere defined), the whole instrument being 
always at the temperature of its bulb. Then the value of one unit of the division 
of the scale, whether this be of approximate degrees or wholly arbitrary, will be 
in true degrees by the mercurial thermometer 

BJ - BJ 
If, then, R" be any observed reading of the thermometer corrected for cali- 
bration and stem exposure, the correction for its reading in ice and for the 
value of the unit of the scale will make this 

R'" = a{R" -RJ). 



If the thermometer is capable of reading in ice but not in steam, then a 
suitable known temperature T h (that of a water-bath, in which the thermometer 
studied and a standard thermometer are immersed together) must be used in 
place of the steam. Then, if RJ is the corrected thermometer reading at T h , 

a= r »-°°. 



THERMOMETRY. 43 

If the thermometer reads neither in ice nor in steam, then two suitable tem- 
peratures, T n and T c , must be used (determined as was T h in the previous case). 
Then 

a= T »- T *. 
RiI-RJ 



For thermometers of arbitrary scale, the stem-exposure correction takes the 
form 

R" = R' + 0.000156 n \_{R' - RJ) a - s°] 

when the thermometer is capable of reading in ice. Note that R", R', n, and 
RJ are all in divisions, but that s° is in degrees, as this would ordinarily be 
taken by means of a cheap thermometer reading in degrees. 

But if the thermometer does not read in ice, then the stem-exposure correc- 
tion takes the form 

R" = R> + 0.000156 n [a <JB' - RJ) + T c - s°] , 
or 

R" = R> + 0.000156 ft [aR' - aR c ' + T c - s°] 

when the temperature — «i? c + T c is a constant for a given thermometer (affected 
only by permanent changes in the volume of the bulb). The letters have the 
same meaning as elsewhere. The derivation of this expression will be seen on 
considering that the direct expression for the correction is 0.000156 n (T° — s°), 
where T° is the temperature of the bulb (which is sought). The approximate 
value of T is a (R' — R f ), when the thermometer reads in ice ; while in the case 
in hand, where R ' cannot be observed, T=a (R' — RJ) + T c , approximately. 

It will be seen that the quantity — aR c ' + T c , if divided by a, would give the 
reading in divisions which the thermometer would have in ice if the scale were 
extended downwards. 

The position of the zero-point is variable. It rises as the thermometer grows 
older, and by continued use of the instrument at high temperatures. It is tempo- 
rarily lowered by sudden cooling ; the amount of the lowering being greater the 
more rapid the cooling and the higher the temperature from which it is cooled. 
These two opposite effects have caused much confusion, owing to the failure to 
notice their independence. Continued heating at any temperature, followed by 
slow cooling, will cause a rise of zero. The amount of rise will be greater as 
the duration of the heating is greater. Thus, if the zero-point be determined 
first, the boiling point then determined, and a repetition of the zero-point deter- 
mination then made, this last will be found greater than before if the thermo- 
meter be cooled exceedingly slowly ; but less than before if cooled rapidly. 

This might be tested as follows (the test will usually be omitted as the time 
required for it is excessive) : 1st. Find zero-point. 2d. Immerse the thermometer 
bulb in a large mass of cold water (several litres), and heat this gradually to 
boiling. Turn out lamp and allow apparatus with thermometer in place to 
cool for some hours, until nearly at the temperature of the air. Remove the 
thermometer and redetermine the zero-point. If the cooling has been sufficiently 
slow, the zero will be found higher than before. 3d. Now heat the bulb of the 
thermometer again in boiling water, and withdraw it while hot, allowing it to 
cool rapidly in the air. Immerse it in ice as soon as cool (this will require less 
than five minutes), aud find zero-point again. It will be found lower than before. 
Repeat this experiment, if the thermometer will admit of it, at some higher tem- 
perature, using an oil-bath. 

The tendency of the bulb of the thermometer is to contract indefinitely, both 
by the action of the cohesion of its molecules, and by the external atmospheric 
pressure, the latter being the less active agent. This contraction takes place 



44 SPECIFIC HEAT. 

much more readily at high than at low temperatures, and is more evident when 
the glass is imperfectly annealed, as after sudden cooling from a high tempera- 
ture. The slowness with which the glass regains the original dimensions after 
sudden cooling is remarkable. The method of annealing glass is governed by 
these facts. 

The zero-point of a thermometer is rendered rather more stable by long con- 
tinued heating at high temperatures. The distance between the freezing and 
boiling points is probably not entirely constant, but becomes less, i.e., the co- 
efficient of expansion of the glass increases slightly with age and use. A new 
thermometer should, before use, be continuously heated for a long time at as 
high a temperature as possible, and then very slowly cooled. The freezing and 
boiling points may then be determined with the same result, whatever the order, 
provided the cooling is not rapid. If this heating has been once carried out, or 
if the thermometer has been much used, the interval from the freezing to the 
boiling point, or the value of 1 division of the scale, will need to be redetermined 
only at long intervals of time ; the change of zero-point, i.e., of i? ' being observed 
by occasional readings in ice. It is usually better to determine the freezing be- 
fore the boiling point for reasons apparent from the above statement. 

4th. Deviation from the Air Thermometer. 

For sufficient reasons the air thermometer is accepted as the basis of tem- 
perature measurement, either directly or with the very slight reduction neces- 
sary to reduce it to accordance with the " absolute scale " of temperature. It 
is essential in most accurate work that the deviations of the mercurial from the 
air thermometer should be known ; but this is only possible by a careful com- 
parison of each mercurial thermometer individually with the air thermometer or 
with a standard mercurial already submitted to this study, for the deviation 
depends upon the rate of variation in the coefficient of expansion of the glass 
at various temperatures. The deviation may amount to as much as 0.5° C, at 
40° 0., 2° to 3° C. at 250°, and 5° to 6° at 300° C. 

The comparison with the air thermometer may be made to aid materially in 
eliminating some other errors of the mercurial thermometers, and must be 
regarded as essential in all accurate research involving any considerable range 
of temperatures. 

SPECIFIC HEAT. 

Find the specific heat of a piece of copper, or other substance, as follows : — 

Weigh the substance when dry. Suspend it from the hook in the cover of 

the heater, inserting the thermometer T (using a high range thermometer, as 

this temperature might burst the calorimeter thermometer) through the hole in 

the cover so that its bulb comes within the substance, if possible. Students who 

have had " Thermometry " will apply the stem-exposure 

correction for T, and will use the thermometer which 

they have calibrated, if it is suitable. Cause the water 

_t7 in AB (about one-third full) to boil gently. Thus the 

Ji\-f: zMB inner tube of the heater becomes surrounded by steam, 



|° which can escape through a side orifice. The tempera- 

ture of this inner space will rise, at first rapidly, later 

1 F ^ G 27 more and more slowly. The substance will have nearly 

the temperature of its surroundings, and, when the rise 

has become very slow, the substance and thermometer may be assumed to have 

the same temperature. This is, of course, precisely true only after the tempera- 



SPECIFIC HEAT. 45 

ture has remained constant for some time, more or less long as the substance is 
a poorer or better conductor. 

While the heating is going on weigh the calorimeter and stirrer C. Fill it two- 
thirds to three-fourths full of water which has been brought to within about 1° 
of the temperature of the room by the addition of hot or cold water, or by 
standing sufficiently long in the room, and weigh. Distilled water is preferable 
but not necessary. Set the calorimeter in place within the outer vessel, and put 
the thermometer and screen in position. 

The quantities which must be known to determine the specific heat are, the 
temperature t s of the substance before its insertion into 0, the temperature t of 
the water in the calorimeter at the moment when the substance is to be inserted, 
the temperature t m of the "mixture" when the substance has acquired the tem- 
perature of the water surrounding it in the calorimeter. As the various opera- 
tions require some time, and the water in the calorimeter is in general at a 
temperature different from that of its surroundings, there is a certain amount 
of heat lost or gained by radiation, and this must be properly determined and 
corrected for. Also the instrumental errors of the thermometers used must be 
properly corrected. Note the number of each of the thermometers used, and 
inquire what the corrections are. These either will be supplied, or must be de- 
termined. 

The order of experiment is this : — 

1st. When the substance is sufficiently heated, and before its removal, insert 
the calorimeter thermometer (dry) between the outer vessel and the calorimeter, 
and against the inner surface of the former, and read carefully the temperature 
to be taken as the temperature of the surroundings t a ; then insert the thermom- 
eter in place in the calorimeter; stir the water thoroughly (stirring must precede 
every thermometric reading) ; read the thermometer every half -minute for five 
minutes, recording the time in minutes and seconds direct, and the observed 
temperature, reading the thermometer to tenths of the smallest divisions. 

2d. Read immediately the temperature of the hot substance, and withdraw the 
thermometer T. Transfer the substance as quickly and carefully as possible to 
the calorimeter, noting the minute and second at which it is immersed. 

3d. Stir the water in C briskly and continuously, reading every half-minute 
the calorimeter thermometer, and recording, as before, the time and correspond- 
ing temperature. Continue the reading for five or ten minutes. 

The observations are now complete. From them we may obtain, — 
W= weight of substance. 
w = weight of water in calorimeter. 

h 

s x = specific heat of substance of which the calorimeter is made. 
s 2 = ditto for stirrer. 
x = specific heat to be determined. 

t s = corrected temperature of substance before immersion. 
t = corrected temperature of water in calorimeter before insertion of substance. 
t m — corrected temperature of water in calorimeter at the time when the sub- 
stance had acquired the temperature of the water. 
t a = corrected temperature of surroundings as found by the calorimeter ther- 
mometer, when dry, and placed against the inner surface of the outer 
vessel, before the insertion of the substance. 
If calorimeter and stirrer are of same material they may be weighed together, 
and, of course, s x = s 2 . If they are of brass or copper, s 1 = s 2 = 0.095. 

To determine t and t m making correction for radiation, the method given in 



46 SPECIFIC HEAT. 

the next paragraph is sufficiently precise. First correct all readings for calibration, 
and all readings which require it for stem-exposure. The reduction to actual 
degrees by the correction for the value of oue division, etc., may be left until 
after the values of t and t m are obtained from the method to be described, and 
these values then corrected, thus saving a considerable number of reductions. 
The process will be exactly the same, of course, with the uncorrected tempera- 
ture readings {i.e., in approximate degrees or in arbitrary units) as for actual 
temperatures which are assumed in the description. 

Cooling Correction. — The temperature of the water at the start must be 
not more than about 1° C. from that of the surrouudings, and in the method 
given in this paragraph it is assumed that this difference is so small as to in- 
troduce no sensible error into the correction if not considered. The com- 
plete correction making allowance for this initial difference between the air 
and calorimeter is given in the succeeding paragraphs. In general, water which 
has been standing in the room for a few hours will be sufficiently near to 
this condition, being, however, slightly cooler than the air. On a sufficiently 
large scale, not less than 0.01 inch to 0.1 of smallest scale-division of the ther- 
mometer, plot points with times as abscissas, and observed corrected temperatures 
of the calorimeter as ordinates. A smooth line drawn through these points will 
have the general form of the line imn" o" p" r", Fig. 28. Find carefully the maxi- 
mum point o" of this curve. The part p" r" of the line Avill be sensibly straight. 
Draw through o" a line o" j parallel to p" r" . Then the length of the intercept 
mj on the ordinate through m, the time of insertion of the hot substance, will 
be the rise of temperature which would have occured in the calorimeter had no 
loss by cooling occurred. The smoothness of the line mo" p" r" will depend 
largely on the thoroughness in stirring the water of the calorimeter, and the 
care and precision in the temperature readings. The line t a is a horizontal line 
whose ordinate is equal to the temperature of the surroundings. The ordinate 
of the point m will give the desired temperature t, and thus the ordinate of j will 
give the desired temperature t m . 

Demonstration. — Suppose, for simplicity, as in the preceding paragraph, 
that t a is so nearly equal to t that the calorimeter is neither gaining nor losing 
heat. This requires that there should be little or no 
evaporation from the surface of the water, otherwise the 
condition of equilibrium is, that the temperature of the 
water is lowered by evaporation to such a point that 
the heat received by it from the surroundiugs is just 
equal to that lost by evaporation, so that the general 
^ statement of this condition should be : suppose that the 

line im is parallel to OX, so that on the whole there is 
neither gain nor loss of heat. Then, during the time t of 
Fig. 28. reaching the maximum temperature t (at o"), the calo- 

rimeter will have been always at a temperature higher than that of the air, and 
there will have been, therefore, a continual loss of heat. This loss will have 
been at an average rate somewhat, but not much, less than the rate shown by 
the curve at p" 1 JI , since the time of heating from m to n" is in general much 
less than that from n" to o". If this latter rate, as shown by the tangent of the 
angle which p" r" makes with OX, be called G, then the loss during the time t 
occupied in the heating from m to o" will be somewhat less than Ct. Now it 
must be noticed that when the maximum o" is reached by the water of the 
calorimeter, the hot substance will still be at a somewhat higher temperature 
than t , since it is still giving out heat to the water at the rate C, for the rate of 
change of temperature of the whole system at o" is zero. The amount of heat 



SPECIFIC HEAT. 47 

yet to be given out by the substance at this temperature is, however, small, and 
it may be shown by a demonstration based on Newton's Law of Cooling that 
this amount is almost precisely equal to the amount by which Cr is greater 
than the true loss by cooling as the temperature of the calorimeter rises from m 
to o". Hence, if no loss by cooling had occurred, the temperature of the 
calorimeter, when the temperature of the substance and water had finally 
become identical, would have been 

t m = t + Cr. 
The graphical method described in the preceding paragraph gives t m by the 
ordinate of the point j. The connection between this demonstration aud that 
graphical solution will be seen by drawing through o" a line parallel to OX, 
aud noting that the intercept on mj between this new line and the point j will be 
equal to Cr, so that the ordinate at j is equal to the ordinate at o" (i.e., t ) 
plus Cr. 

This, however, is only for the case when the initial rate of cooling as indi- 
cated by im is zero, and the results of this method are more satisfactory the more 
closely that condition is fulfilled. The complete correction when C is not zero 
will now be given. Suppose the line t a parallel to OX to correspond to the 
temperature at which C would be zero, and suppose that there is no evaporation, 
so that t a is the observed temperature of the surroundings. Then, if t a does not 
differ much (say more than one-third of mj) from t, the following demonstration 
and method will be quite satisfactory. Suppose t a to be between t and t in the 
present case, then im will incline upwards to m, and the calorimeter will receive 
heat up to the time when its temperature reaches t a . Let C be the tangent of 
im with OX, as found from the plot or computed from the observation ; then, 
until t a is reached, the calorimeter gains heat from its surroundings at an average 
rate (since h" m is sensibly straight, and the gain or loss of heat is proportional 

C 
to the difference of temperature) of very nearly — , and thus gains a total amount 

c 2 

— hh". Moreover, the correction Cr as previously deduced on the supposition 

that t = t a makes allowance for a cooling during this time when there is now a 
heating, and a complete demonstration would show that this allowance was at 

C C 

very nearly this average rate — , and in total amount nearly — • hh" . Hence, 

the total correction deduced for the former case, viz., Cr, is too great for the 

C 
present case by 2X — hh", and the desired temperature would now be 

t m = t + Cr-C'-hh". 

It should here be noted that the signs of the rates C and C are taken as + 
when they correspond to a cooling, and — to a heating from without. 

In the graphical solution of the complete correction, Fig. 29, this value of t m 
would be obtained by subtracting from j the quautity hw, which is the intercept 
on the ordinate mj, between the line t a , parallel to OX, and the line h"wi\ drawn 
through h" parallel to mi. Or t m might be obtained by dropping 
from h" the ordinate uutil it passed at x through the line im pro- 
longed, and then drawing the line xu perpendicularly to mh ; when 
uj = t m — t, or t m =t-\- uj. If t a is below m, the term C- hh" becomes 
+ instead of — , and in the graphical solution the line im will incline 
downwards towards m. It will be seen that so long as t a — t is 
small in proportion to t — t the term C'-hh" remains very small, 
and may be dropped entirely, thus leading to the first approximate 
method given when C'-hh" does not exceed about ten per cent of Cr, as the 
value of the latter cannot usually be determined within that degree of precision. 



48 



LATENT HEAT OF VAPORIZATION. 



This method of correction is derived from one first given by Professor Row- 
land. 

Specific Heat of Calorimeter. — The specific heat or water-equivalent of 
the calorimeter may be determined by the following method with a fair degree 
of precision. Of course the better way is to know the specific heat of the 
various materials of the calorimeter and the weights of those materials, or to 
determine the specific heat of a mass made up of those materials in the proper 
proportion. 

Insert in the calorimeter enough water at about 5° above the atmospheric 
temperature to somewhat less than half fill it. Stir continuously, and note time 
and temperature each half minute for about five minutes. In a beaker have 
a weighed amount of water at about the temperature of the air, sufficient to 
somewhat less than half fill the calorimeter. Note the temperature of this 
water by another calorimetric thermometer from minute to minute, before 
taking that of the warm water as just described. At a noted time pour the cold 
water carefully into the warm, stirring and reading temperature of the mixture 
at half-minute intervals. 

Plot the temperature observations as in the specific heat measurements, and 
find the temperature of both warm and cold water at time of mixture, and maxi- 
mum temperature after mixture. The latter may be found by extending back- 
ward the line of fall of temperature after mixture to the time of mixture, since 
the rise of temperature was by so sudden a jump, in this case, that the loss by 
cooling was substantially all at, or very near the maximum temperature of the 
mixture. Assuming the specific heat of water to be unity, or taking the value 
of the specific heat from the results of the work of Regnault or Rowland, the 
equation can be readily deduced, and the specific heat of the calorimeter com- 
puted. 



LATENT HEAT OP VAPORIZATION. 



When a liquid at t° is converted into a vapor at the same temperature t°, and 
under a constant pressure p, a definite quantity of heat per unit weight of liquid 

is taken up by the liquid. This heat becomes 
partly potential energy, by the change of relative 
position of the particles of the substance in the 

\[ji ^j-^- changed state of aggregation, and is partly ex- 
-^Yi pended in work external to the mass of the liquid, 

i.e., in the work of pushing back the atmospheric 
pressure p through the space corresponding to 
the increase of volume of the water expanding 
into steam. Let v x be the volume of water at t°, 
and v 2 that of the resulting steam at t° ; then the 
work done against p will be p (v 2 — v x ). If the 
atmospheric pressure be 760 mm., so that water boils at 100° C, the total number 
of thermal units (kgrs. of water raised 1° C.) required to evaporate 1 kgr. of 
water will be 536.5; of this amount 40.2 units will be expended in work against 
atmospheric pressure, and hence 536.5 — 40.2 = 496.3 will remain as energy of 
some sort, chiefly potential energy, in the vapor, or 92.5 and 7.5 per cent respec- 
tively. The quantity 536.5 is called latent heat of vaporization, and may be 
measured as in this experiment. 

The steam generated in the heater AB passes into the condenser C which is 
open to the air. This condenser is surrounded by a known weight of water in 




Fig. 30. 



LATENT HEAT OF VAPORIZATION. 49 

the calorimeter, and the rise of temperature of this is noted by the thermometer T. 
The increase of weight of the condenser gives the weight of steam condensed. 

L= Latent heat of steam. (Sought.) 
c= weight of condenser dry. 
w = weight of water in the calorimeter. 
w' = weight of calorimeter. 
w" — weight of stirrer. 
s c , s', s" = specific heats of condenser, calorimeter, and stirrer, respectively. 
If of copper or brass, as in this case, sp. ht. = 0.095. 
W= weight of steam condensed. 

Z c = corrected temp, of water just before passing the steam into C. 
t n = corrected temp, of water just after steam is cut off. 
t s = computed temp, of steam. 
H= height of barometer in mm. reduced to 0° C. 



The calorific capacity or water-equivalent of the calorimeter and contents 

willbe w + w'sr+w"s»+cs e ; 

or, if w', w", and c are all of copper or brass, 

w + 0.095 (w'+v/'+c). 

The rise of temperature of this mass of material is % — t c . The product of these 
two quantities will be the gain of heat, in thermal units, by the calorimeter and 
contents ; and this must be equal to the latent heat of the weight of steam con- 
densed, plus the heat given out by the condensed water in cooling from the 
temperature of the steam t 8 to the temperature t h of the water, i.e., equal to 

WL + W(t s -t h ). 

The temperature t s may be determined from suitable tables (see Smithsonian 
Collection of Tables, Landolt's " Physicalisch-Chemische Tabellen," or Wullner's 
"Lehrbuch der Experimental-Physik " vol. iii.), or may be computed by the^ 
approximate formula 

£ s =100° + - OS- 760). 
80 

Hence, WL + W (t s - t h ) = (w + w's'+ w"s"+ cs c ) (t h - t c ) , 

or WL + W(t s - t n ) = (w + 0.095 [>'+ w"+ c])(t h - t c ), 

from which L may be computed, all the other quantities being known. 

To avoid, as far as possible, the introduction of steam containing condensed 
water, i.e., " moist" or " primed" steam, the boiler is arranged, as shown in the 
figure, with sloping plates extending partly across its section, intended to arrest 
particles of water spattered from the boiling surface. The tube leading to the box 
EF is steam jacketed, as shown, so that the steam to be used passes only through 
the inner tube. The box EF serves as a steam chest, and steam is taken from it 
for the condenser through a cock operated by a rod through the side of the box. 
The entrance to the cock is downwards, so that any water condensed within it 
while the cock is closed may return into EF and not pass into the condenser. 
The connecting tube between this and the condenser is within the steam, except 
a short projection to which the rubber connector is attached. The outside of 
the whole steam space is wrapped with felting, as far as possible, to reduce loss 
of heat and consequent condensation and possible moistening of the steam. An 
exit tube is provided at E, which, ordinarily, will not be needed, and should be 
closed. A tube at D serves to drain away the water condensed in EF, and to 



50 LATENT HEAT OF VAPORIZATION. 

furnish an escape for the steam which does not find its way into the condenser 
during any part of the experiment. From D the water and steam should be led 
away by a rubber tube terminating in the air within some suitable receptacle. 
Too much resistance in the outlet will cause the water in AB to boil out through 
the side opening at A ; but the resistance at D must not be too small, as, in such 
case, little steam would flow into the condenser. 

Eor the experiment, disconnect the condenser, and remove the' calorimetric 
apparatus to such a position that the steam escaping from the tubes will not 
affect it. Start the Bunsen burner beneath the boiler AB, opening the cock in 
E, and allow the steam to issue through the rubber tube which' is to connect 
with the condenser, and through the orifice D, which is to remain open through- 
out the experiment. Thorough heating of the apparatus before the beginning of 
the measurements is thus assured. Be careful that the water is not wholly 
evaporated from the boiler. 

In the meantime, weigh the calorimeter and stirrer when dry. Fill about 
three-fourths with water, weigh again, and subtract to get the weight w of water 
used. If convenient, a suitable known weight of water may be readily intro- 
duced by means of a flask holding, when filled to a reference mark on the neck, 
the desired amount of water — a rapid method, precise to 0.5 per cent., or less, 
if carefully employed. Empty the condenser of water, as far as convenient, dry 
its outside surface, and weigh it to miligrammes. Its weight, when thoroughly 
dry, is marked upon it, or may be obtained from the instructor. The excess of 
the observed weight above that when dry is due to the water remaining within, 
which cannot be quickly removed. Add this excess to the weight of the water 
in the calorimeter, for it plays the same part as if it were a portion of that 
mass. 

When the calorimeter is properly filled with water, close the cock in EF, 
continuing the boiling and allowing the steam to escape through D. Set up the 
screens SS. Slip on to the upper end of the condenser coil a piece of rubber 
tubing of such length as to extend beneath the surface of the water in the calo- 
rimeter when the condenser is in place. This will largely prevent the loss of 
heat in the calorimeter by evaporation of water where the hot part of the coil 
passes through the surface. Leave, however, about half an inch of the upper 
end of the metal coil free. Insert this into the rubber connector on the end of 
the tube from the cock in EF, having the condenser in place within the calorim- 
eter. Put the thermometer carefully in position. Stir the water in the calorim- 
eter continuously, and take half-minute readings of the thermometer for five or 
ten minutes, recording times and readings as in the experiment on " Specific 
Heat." At a noted time open the cock. Record temperature each half minute 
or minute, stirring continuously until the temperature of the water has risen 
about 4° or 5°. Then, at a noted time, close the cock, and continue half-minute 
temperature readings for ten minutes. Observe the temperature of the air by 
the calorimeter thermometer (dry) placed with its bulb inside the outer shield 
of the calorimeter; and read the barometer and attached thermometer. The 
temperature observations are to be treated exactly as in the experiment on 
"Specific Heat" to obtain the corrected rise of temperature (t h — t c ) of the 
calorimeter. Finally, detach the condenser, dry its exterior, and weigh. The 
gain in weight will 1oe the weight of the steam condensed = W. Compute L from 
these data. 



EXPANSION OF LIQUIDS. 51 



EXPANSION OP LIQUIDS. 

• 
The discussion of Coefficients of Expansion and of the relation betweeu the 
true and mean coefficient is given in the lecture notes on that subject. These 
should be reviewed in this connection. The mean coefficient of expansion of a 
liquid between 0° and t° is the average fractional increase in volume per degree 
from 0° to t°, the fraction being expressed in terms of the volume at 0°. 
Let 

v = the volume of the liquid at 0°, and 

v t = " " " " " t°, some measured temperature; then 

the fractional increment of volume in this temperature interval of t° will be 



v 
and t° will be given by 



given by 1 Z, and the mean coefficient of expansion of the liquid between 0° 

j given by 



v t 



The true coefficient at t° is the rate of expansion at that temperature. Eor further 
definition and the relation to the mean coefficient, refer to the lecture notes. 

Of the various methods for obtaining the coefficient of expansion of liquids, 
the following is usually the most convenient when sufficient quantities of the 
liquids are obtainable. It has the advantages of being direct, and capable of a 
considerable degree of precision. 

A light flask of the shape shown in the sketch is to be used. The stopper, 
consisting of a closed glass tube, ground into place, serves to avoid loss 
by evaporation during cooling and weighing. A reference mark is placed 
upon the narrow part a of the neck of the flask. It is evident that if the flask 
be weighed when containing successively enough liquid to fill it to the reference 
mark at various measured temperatures, coefficients of expansion cor- 
responding to those temperatures may be determined. Let 

w = weight of flask empty, 

w — " " " containing enough liquid to fill it at 0°, 

w t = " " u " " " " " t°, 

&=::mean coefficient of expansion of liquid between 0° and t°, 

Fig. 31. 
k t = " " " cubical expan. of glass between 0° and tP. 

Eor most liquids /3 varies widely with the temperature. The coefficient for glass, 
however, varies but little with the temperature, and may be assumed as constant 
and equal to 0.000026, unless the coefficient for the kind of glass used is known. 
To deduce the value of £ from these data, call s the specific gravity of the liquid 
at 0°. This need not be known, as it will be eliminated from the final equation. 
The volume of the liquid filling the flask at 0°, and hence, of course, the volume 
of the flask itself at 0°, is w °" w . The liquid filling the flask at t°, and weighing 

w t -w, would, when cooled to 0°, have a volume !^Z^. This volume of liquid, 

when heated to t°, would increase to w t~ w (i + pj) f an( i would just fill the flask 

s 

which at t° would have a volume w °~ w (i + k t t) . Hence, 

s 




52 EXPANSION OF LIQUIDS. 

I + fat _w — w 
1 + kt ~ w t — w 

whence fa may be computed. Where k is not known from direct measurement 
(e.g., by measuring the apparent expansion of mercury in the flask), the com- 
putation will be sufficiently precise if the approximation- — =^- = 1+ (& — &)£ 

1 4- kt . 
is used, thus reducing the expression to 

w t — w 
whence fa= ™ ~ w < fc 

By making a sufficient number of weighings corresponding to w t and of 
measurements of t, three or more values of fa may be computed. From these, 
by the method of least squares or otherwise, may be deduced the constants fa, 
A, B, etc., in the equation 

fa=fa + At+Bt 2 + etc., 

which will then give the means of readily computing at any time the value of fa, 
the mean coefficient of expansion from 0° to t°, for any desired temperature t. 
Then, from the known algebraic relation between the mean and true coeffi- 
cients, the equation for the latter can be deduced, since 

y t =fa+2At+BBt 2 + etc. 

The constants for both of these equations are to be determined for some liquid ; 
and, for the method of computation by the method of least squares, the student 
is referred to the notes on that subject. 

To obtain the weights w t , w , etc. (to mgr.) and the corresponding temper- 
atures with sufficient precision, very careful manipulation is necessary. First 
weigh the flask dry. Then boil the liquid for some minutes, or heat it to a tem- 
perature above that at which it is to be used, and stir thoroughly. This serves 
to remove the air in solution, which would otherwise interfere with the precision 
of the result. If this method of removing the air is not admissible, it may be 
done by placing the liquid under the air-pump and exhausting. , Most liquids take 
up air with considerable readiness, and it is therefore necessary to prevent, as far 
as possible, any change in the amount of air absorbed during the whole of the 
measurements. Fill the flask with the liquid at the temperature of the room, 
leaving the liquid well above the reference mark. When the neck at a is quite 
small, as ordinarily it should be, it is sometimes troublesome to fill the flask. In 
such a case, draw out a glass tube to a fine long point, insert this point clown- 
ward through a into the flask, pour the liquid into the upper part of the neck, 
and exhaust the air from within the flask through the glass tube by the mouth, 
or by an aspirator. This process is much quicker than the ordinary one of 
heating and cooling alternately, and is preferable to putting the whole apparatus 
under an air-pump. Next surround the flask by flnely-pouncled ice, as in testing 
thermometers, and allow the liquid to cool to 0°, maintaining the level of the 
liquid above a for some minutes until thoroughly cooled. Then, by means of a 
fine tube, filter paper, or otherwise, draw the liquid down to the level of a while 
in the ice. Observe, for some minutes, to see that no further contraction takes 
place ; and, when that point is reached, dry, as thoroughly as possible, the 
inside of the neck of the flask. Stopper the flask. Remove it from the ice, and 
allow it to stand in the air, or put it into a water bath at a temperature slightly 
above that of the air, until it is warm enough to weigh. Then dry thoroughly, 
and weigh to miligrammes, obtaining thus w Q . To obtain w t at any temperature 



EXPANSION OF GASES. 



53 



below 100° C, a double water bath may be used. This consists of two cylin- 
drical vessels, a smaller within a larger one. At temperature above that of the 
air, the outer bath may be maintained at any nearly constant temperature desired 
by means of a flame, and the inner temperature will thus be still more constant. 
The liquid in the flask should be drawn down to a after the inner temperature 
has remained as nearly constant as possible for five minutes or more, and the 
temperature carefully determined, using all the necessary corrections. For tem- 
peratures of very nearly 100°, the flask may be placed within steam, as in testing 
thermometers, and the actual temperature may be computed from the corrected 
barometer reading at the time. Observations of w t should be made for at least 
three temperatures besides that of 0° if the equation is to contain the term Bt 2 , 
and for at least four places if the next term Ct s is to be retained, and so on. 
An additional observation will serve to determine k if necessary, though this is 
seldom done; but, of course, the precision of the constants is increased by 
taking more numerous measurements. It will be seen, upon consideration, that 
since the liquid, when weighed, is always at very nearly the same temperature, 
the corrections for displaced air affect the different weights in proportion to those 
weights, except in so far as the density of the air varies ; and, as £ is computed 
from a ratio of these weights, and not from absolute weights, all these correc- 
tions are eliminated, except in cases where the variation of density of the air 
would affect the result beyond the limit of precision otherwise attainable, which 
rarely occurs. 

EXPANSION OP GASES. 



6 



ft 



II 



The apparatus used for this experiment is a convenient form of air thermom- 
eter. A glass bulb B is connected by capillary glass tubing (" barometer tubing ") 
with a larger tube at C, which is connected through a stop- 
cock D and a rubber tube to an open tube A of the same 
diameter as C. Both A and C are adjustable in height upon 
the vertical support by clamp screws. The expansion of 
the air in B is measured indirectly by the change of its pres- > * < '*< J 
sure when its volume is maintained constant, and the bulb 
changes temperature. To measure the coefficient of expan- 
sion of the air, the bulb is first immersed in ice and afterward 
in steam. 

The three-way stop-cock at D may be understood from 
Fig. 33. Nos. 1 and 2 show the vertical section and the 
front of the cock in the same closed position. There are four 
possible closed positions, viz., as shown in 1, and 90°, 180°, and 
270° from this position. The flattened or marked side of the head of the stop- 
cock is on the same side as the lateral boring of 
the cock, and the handle of the cock is parallel 
with the through boring. Whenever the handle 
or rod is at an angle of 45° to the horizontal all 
borings are closed. Position 3 shows the only 
proper position to connect the bulb B with the 
mercury column AD. The cock should never 
be left in this position, but should always be FlG# 33# 

closed, except when the mercury is in process 
of adjustment. 

The air in the bulb must be thoroughly dried. This is facilitated by having 
a point, or fine tube, left at the lower side of B when this is made. The process 



Fig. 32. 




54 EXPANSION OF GASES. 

of filling would then be to connect this fine tube, by a rubber tube, with the 
drying apparatus containing fused calcic chloride and sulphuric acid, and to 
connect the side opening of I), in position 4, with an aspirator or air-pump. 
The bulb B should be kept at about 100°, and dry air drawn slowly through it 
for some minutes. The bulb should then be allowed to cool with D closed but 
the fine tube open ; and, when at about the air temperature, the fine tube should 
be sealed in a flame, and the mercury, formerly standing just at D, should be 
forced up into C, and the cock closed. If B is not provided with such a tube, 
it must be filled by repeatedly exhausting the air from it while in steam, and 
allowing dry air to enter. This may readily be done by connecting the side 
opening of D with a glass T, one arm of which goes to an aspirator or pump, 
and the other to the drying tubes. A pinch cock on the rubber connectors of 
each branch serves to shut off the drying tubes while the exhaustion is going on, 
aud to shut off the pump while the dry air is allowed to enter B. Care must be 
taken to introduce no dust with the air. After repeated exhaustions, the final 
filling of B should take place, with the bulb at about the air temperature, and the 
mercury should then be forced up into C. Whenever the apparatus is left with 
D closed, the column at A should be high enough to make the pressure outward 
at £>, and thus to avoid any introduction of air by leakage. 

1st. Place in the apparatus provided for this purpose a layer of finely-broken 
ice. Then place this upon the supporting stand at a suitable height beneath 
B, and lower C into place, having D closed throughout this and all other similar 
manipulations. Do not attempt to adjust the ice-box up to B, but always lower 
C into place. Place in position the screen between the box and the mercury 
columns. Carefully cover the bulb, to the depth of an inch at least, wholly with 
fine ice, covering the' top of the ice with a flannel, or other cloth, to prevent air 
currents, etc. In about five minutes, the air in B will have attained the tempera- 
ture of the ice. Then lower A until the mercury surface is about on a level 
with the top of C; open D slightly, and observe whether the mercury in Crises 
or falls. Move A either up or down, as may be necessary, using the vertical 
adjusting screw for the final setting, until the top of the mercury meniscus is 
exactly opposite the reference mark, near the top of C ; the cock being open, of 
course, during this adjustment, so that the air pressure within the bulb is exactly 
counterbalanced by the atmospheric pressure and difference of level of the mer- 
cury columns A and C without. 

When this adjustment is finished, close the cock D, and take the readings of 
the columns at J. and at C by whatever means is provided with the apparatus used. 
For readings closer than 0.1 mm. a cathetometer must be used. Read the tem- 
perature of the thermometer hanging upon the apparatus, which will give approx- 
imately the temperature of this mercury column and of the small volume of air 
above (7; call this t. Reduce the observed difference of level of A— C to the 

freezing point ( \ = - — - rrrrrr-. ), and call the reduced difference h v This as- 
^ 1 -J- O.OOOlolf / 

sumes the scale used to be correct at about the temperature of the air. If the 
cathetometer be used, the instrumental errors and coefficient of expansion to be 
used must be requested from the instructor. Read the height of the mercurial 
barometer and the temperature of the attached thermometer. Reduce this 
height to the freezing point by the tables, and call the reduced reading H v The 
temperature of the air in the tube above C is tj°, while that in B is 0°. The 
correction for this difference is as follows. Let v (obtained as described further 
on) represent the volume of the air above C at ^°, then its volume at 0° would 

be , where o is the coefficient of expansion of air. This air is, of course, at 

1+ ot x 

at a pressure of H x + h v as is also the air in the bulb. 



EXPANSION OF GASES. 55 

Care must be taken that the mercury never falls below the lower part of the 
tube C, and that it is not forced over into the bulb. If the former occurs, the 
experiment must be repeated, owing to the loss of air; if the latter happens, ask 
for instructions. The cock D must be closed at all times, except when the final 
adjustments are being made. Whenever the instrument is to be left, D must be 
closed; the column at A should be high enough to make the pressure at Z> out- 
ward, and thus avoid introduction of air. The meniscus in C should also be 
drawn down to a point below the reference mark, so that any soiling of the tube 
during long standing will not be at the reference mark. 

2d. This portion of the experiment consists in measuring the pressure of 
the air when the bulb is at the temperature of the steam from water boiling 
under a measured atmospheric pressure. Having carefully removed the ice 
from around P, raise the bulb, remove the ice-box and stand, and replace them 
by the copper heater, containing about an inch depth of water and resting upon 
the gas-stove. Or if the heater and ice-box are combined, light the gas beneath 
the apparatus, and allow the ice to melt as the heating progresses. In placing 
the cover upoL the heater, put the marks upon cover and heater together, as 
they fit loosely in this position. As soon as the water begins to boil, raise A 
about nine inches ; then open D, and adjust the mercury to C as before. Read 
again the height of C and A, and note the thermometer, which now indicates t 2 . 
Read also the barometer and attached thermometer. 

In careful measurements with the cathetometer, it is important that a certain 
sequence should be used in the readings of the barometer C, A, etc., in all cases. 
For this, inquire of the instructor. 

Close D, and turn off the gas. Call H 2 the corrected barometer reading, and 
7i 2 the corrected difference of level of A — C. Then H 2 + h 2 is the observed pres- 
sure of the air in the bulb. The volume of the- air above G is now, as before, v ; 
the slight expansion of the glass from t 1 ° to t 2 ° being negligible in this correc- 
ts 

tion, but its temperature is L, so that its volume at t x ° would be — . The vol- 

1 + at 2 

ume of the bulb at 0° is F ; but at T° (=the temperature of the steam) it is 

V (1 + kT) when k is the coefficient of expansion of the glass (= 0.000026). This 

1 + kT 
volume of air reduced to 0° would, therefore, be V — — ™ # Thus the total vol- 

+ 14-kT 
ume of the air in the bulb and above C reduced to 0° would be F ~ - + 

1 + al 

V 

— — 7f under a pressure of H 2 + h 2 . 

1 + al 

Now, from Mariotte's Law, we know that for the same mass of gas, at a con- 
stant temperature, the product of the. pressure by the volume is a constant. 
Hence, the product of the volume of the air reduced to 0° by its pressure in each 
of the above cases, since the total mass of the air is unchanged, should be the 
same, and we have, — 



^+rf^W^^ + rfJ^ 



); 



or, dividing both sides by V 



In this equation, the only unknown quantity is a, which may, therefore, be 

calculated. To facilitate the computation, the approximate value a =0.003670 

v 1 v 1 

may be substituted in the term — • — and — • , these being small 

V o 1 + at^ Vo 1 + of, 



56 EXPANSION OF GASES. 

corrections only, and a value of a thus deduced. Should this value of a differ 
by 1 part in 500 from the assumed value, substitute it in place of the assumed 
value, in the two terms referred to, and recalculate a, thus obtaining a more 
correct value, one recalculation is ordinarily sufficient. 

The temperature T of the steam may be ascertained as stated at p. 49 of these 
notes. 

When the cathetometer is used, and the greatest attainable precision is sought, 
correction must be applied to this value of T for the change in barometric height 
by the variation in the force of gravity with the latitude and elevation above 
the sea. 

To use this instrument as an air thermometer, it is necessary first to determine, 
once for all, a as just described; then immerse the bulb in any bath of which the 
temperature is to be measured ; measure A — G and the barometric height, and 
substituting H and h as thus found in above formula, compute the value of T. 

3d. The value of Fand v are to be obtained as follows if they are not required 
with a precision of more than 1 part in 50. 

Obtain by measurements with a string, tape, or otherwise, as nearly as pos- 
sible, the average circumference of the bulb. Allow about 0.5 mm. for thickness 
of the glass, and compute the capacity Fin cc. When possible, it is, of course, 
best to determine Fby weighing the bulb filled with water. To obtain v, turn D 
into the position 5; raise A somewhat above C; open D slightly (position 3), so 
that the mercury may gradually flow into the capillary CE, and let it flow to the 
point E about where it was immersed in steam. The capillary is so small that a 
difference of some inches in this position will not introduce much error into v. A 
careful adjustment is necessary only in the most precise investigations. Then 
turn D into position 4, holding beneath the side-opening a small beaker to take 
the outflowing mercury. Draw off the mercury until the meniscus is as in the 
measurements, just at C. The volume of mercury drawn off must then be v. 

„ T . , ., T , , .,, „ , wt. wt. in grammes 

Weigh the mercury. Its volume will, of course, be = ~~ • 

° J Sp.gr. 13.6 

Repeat this determination of v, and average the two results. 

Care must be used iu the computations of the results that the proper number 
of places of significant figures is used at all points in the work, so that the com- 
putation shall not vitiate the results. With the simplest form of apparatus used 
in the laboratory, results should be obtained within 0.5 per cent of the true 
value, which is 0.003670. If the cathetometer is used, and the air thoroughly 
dried, the results should be within 0.2 per cent. With imperfectly dried air, a 
considerably greater value of a is obtained, even up to one per cent too large. 

When any considerable number of computations of either a or T are to be 
made, much saving of labor is effected by using the following simplified formula, 
and throwing the work into systematic form. Both a and T should be com- 
puted from the same formula. 

H-h' + ^H.- 



aT _ V 1 + 7*8 



V 1 + tf 

where y = a — k = 0.003644, 

h = H 1 + h l , 



a 



V_ _±_\ 



Of course, when the greatest precision is sought, the values of k must be de- 
termined for the particular bulb and for all temperatures. 



ADJUSTMENTS OF THE CATHETOMETER. 57 



ADJUSTMENTS OF THE CATHETOMETER. 

The following adjustments include the most important details in the use of 
the Stauclinger Cathetometer. They are best made in the sequence here indi- 
cated, although in some cases this order is not essential. Thus adjustments 
7 and 8 can be made at any time independently of others, and 7 requires 
frequent attention. Adjustments 1, 2, 3, and 8 should never be disturbed unless 
found to be imperfectly made. The student should note carefully the points 
which should be attended to at each use of the instrument, and be sure that 
they are never neglected, as injury to the instrument, or imperfect results would 
be sure to follow. Adjustments 1 and 2 once rightly made require no further 
attention, and are, from their nature, performed when the instrument is first 
used after being set up. Adjustment 3 requires attention only when the slide is 
seen to be slightly loose, or much too tight, but the bar must be carefully wiped 
and oiled before use on each separate clay of use. In every motion of the slide 
up or down, the vernier must be turned away from the scale lest it wear a groove, 
and thus lessen the precision of reading. Adjustment 4 once made remains 
right unless disturbed in other manipulation. It must be occasionally tested. 
Adjustment 5 must be tested at least as often as at the beginning of each day's 
work, and 6 is made at the same time. Adjustment 7 must be made whenever 
the cross-wires are not in distinct focus. Adjustment 8 once properly made 
requires no further attention except at the beginning of any important work. 

1. On the central steel shaft of the instrument are two conical bearings, one 
near its top, the other near its foot. The hollow brass sleeve carrying the 
rotating portions of the instrument has two corresponding conical surfaces 
within, and thus, when in place, settles down firmly upon the bearings of the 
steel shaft. These make excellent guides for the rotation, but, as will be 
readily seen, the friction is great. To obviate this and still retain the advantage 
of the guides, the steel axis is finished at the top as a flat cone. The apex of 
this is pressed upon by a steel plate, the amount of the pressure being regulated 
by means of a screw turning through a nut in the top of the brass sleeve in the 
line of the axis. By sufficiently turning this screw downward, the whole weight 
of the rotating parts may be borne by the plate, and the friction on the guides 
thus removed. The best effect is reached, however, by relieving the cones of 
but part of the weight, adjusting the screw so that the freedom of rotation is a 
little short of complete, a good bearing being thus insured. 

2. The bar carrying the graduated scale, and upon which the telescope and its 
attachments receive their vertical motion, should be very nearly parallel to the 
vertical axis of rotation. The only error arising from want of such parallelism 
is the apparent lengthening of distances measured ; for it is obvious that, were 
the scale tipped from the vertical through an angle a, a vertical distance, found 
by measurement with the inclined bar to be I, would, in reality, be I cos a. This 
adjustment is really made, at first approximately, by judgment by the eye alone. 
Then, after the later adjustments, this can be finally made with sufficient close- 
ness by a carpenter's level or a plumb-bob. 

3. The slide carrying the telescope and attachment must run easily on the 
bar without being loose. This adjustment, which is somewhat difficult, is made 
by means of a pair of screws on one side of the bar, which carry the portion of 
the slide bearing upon this edge of the bar. The least play of the slide allows 
change of level of telescope, and thus inconvenience and error. The bar should 
be thoroughly wiped and oiled on each day of use. 

4. The adjustment of level so that it shall be parallel to the axis of the tele- 



58 ADJUSTMENTS OF THE CATHETOMETER. 

scope is necessary to facilitate subsequent adjustments. Place the level in 
its proper position on the telescope tube. Read the position of the bubble. 
Then turn the level end for end, and read again. If the level be in adjustment, 
the end of the bubble towards (say) the object glass of the telescope will each 
time be at the same distance from the centre line of the level. If these two 
readings are not the same, the adjusting screw provided at the end of the level 
must be turned until the condition is fulfilled. 

The explanation is this. Let e/be the direction of the axis of the telescope, 

which may or may not be horizontal. Let cd be the direction of the surface at 

the middle point of the level when adjusted parallel of ef. In' this case, the 

bubble will stand nearer the higher end. Now, on 

<^ 3 _~~^a_ j.- z~J~ a reversing cd, c will, of course, be brought to d, and d 

e „ ? to c ■ ; the inclination of the surface of the level will 

Fig. 34. 

therefore be unchanged. Hence the end of the bubble 
will still stand at the same distance from the middle, and toward the higher end. 
If, however, the original position of cd had been gh, inclined to the proper line 
cd, then, on reversing, it would have moved to g'Ji', and the bubble would have 
changed position. 

By turning the telescope into parallelism with two of the levelling screws of 
the base, clamping the vertical axis, and then approximately levelling by means 
of these screws, this operation is facilitated. 

5. Rendering the axis of the telescope accurately perpendicular to the vertical 
axis of the cathetometer. 

First render the axis approximately vertical, by going through once with 
adjustment 6. Then read the position of the bubble. Turn the instrument 
about its vertical axis as nearly as possible 180°. If the telescope be perpen- 
dicular to this axis, the bubble, if not exactly in the middle of the tube, will 
shift to a second position symmetrical with the first with regard to the middle 
point of the level. In other words, if, in the first position, the reading of the 
end of the bubble towards the highest end of the tube be taken, then, after the 
180° revolution, the reading of the end of the bubble towards the highest end of 
the telescope will be the same as before, but it will be the other end of both 
bubble and telescope. For, if mn represent the axis of the instrument, — not 
necessarily vertical as yet, — and cd the direction of the middle of the surface 
of the level (which has been rendered parallel to the axis of the telescope) when 
perpendicular to mn ; and, if a be the position of the end of the 
bubble toward the high end of the telescope ; then, as cd is 
perpendicular to the axis mn, if rotated 180° around that axis it 
will come into the same direction, but with c brought to d, and 
ft d to c. Hence the middle surface of the level will have the 

J* same direction, and the opposite end of the bubble will now 

Fig. 35. come to the same distance from mn as before, but now a will 

be toward the opposite end d of the telescope. It will be readily 
seen, on inspection, that this will not hold true when cd is not perpendicular 
to mn. 

Inasmuch, however, as it is difficult to turn cd through exactly 180° by the 
eye, and as mn cannot yet be rendered exactly vertical, it is evident that in the 
second position of cd a slight difference in position of a may arise, in virtue of 
these causes combined, even where cd is exactly perpendicular to mn. The 
adjustment by this method is, therefore, an extremely tedious one, and the 
following is to be preferred. 

The whole instrument rests on three levelling screws, and by these it should 
be approximately levelled by performing 6 once. Turn the instrument around its 



■m 

; -=4 — * 



ADJUSTMENTS OF THE CATHETOMETER. 59 

vertical axis until the telescope is in line with one pair, which may be called I. 
and II. of these screws. Lower the micrometer thumb-screw under the end of the 
telescope (in use this adjusting screw should always be at the eye end of the tele- 
scope) , so that the round-headed screw shall rest on the middle of the movable 
steel plate provided for it. Adjust this screw until the middle of the bubble is at 
the middle of the level. Turn the instrument until the telescope stands over screws 
II. and III. Now turn screw III. until the bubble is again in the middle. Revolve 
the instrument, until the telescope stands over screws III. and I., and find how 
many divisions the bubble is away from its middle position. If it still remains in 
the middle, cd is perpendicular to mn. If the difference from the middle is p divi- 
sions, then the round-headed adjusting screw under the eye-piece is to be turned 
until this difference becomes %p, when the adjustment is approximately correct. 
If, however, one end of the bubble is concealed, or is very near one end of the 
tube, the necessary change in the adjusting screw may be considerably less than 
that corresponding to Jjp, owing to the rapid change of curvature of the tube of 
the level near the end. A repetition of the whole operation, beginning with the 
telescope over I. and II. , levelling by II. until bubble comes to middle ; turning 
to II. and III., aud levelling by III. ; then turning to III. and I., and reading the 
bubble again, will either show the former adjustment to be accurate or enable 
the observer to improve it. Two or three repetitions in this way will give a 
correct adjustment, and it is to be observed that, by this operation, the central 
axis mn incidentally becomes vertical, thus rendering 6 unnecessary. The 
reason for this halving of p will be readily seen on considering that if in the 
first position, I. to II., the line cd have the end c over I. and d over II., and if cd 
instead of being perpendicular to mn have the end d a little too high, then on 
adjusting II. until the bubble comes to the middle, the foot at II. is necessarily 
turned too low by a certain small amount. In the same way in the position II. 
to III., the foot at III. is turned below that at II. by an equal amount. So that 
when the position III. to I. is reached, the foot at I. is found higher than the 
foot at III. by the sum of these two equal quantities. Therefore, halving the 
inclination of the level in this point, which is approximately done by moving 
the bubble back to %p, will make the line cd nearly or quite perpendicular to mn. 
This adjustment must be tested on each different day of use of the cathetometer. 
Moreover, as the slide bar can never be perfectly true, the adjustment will not 
be exactly the same for all parts of the bar. This error is nearly eliminated by 
the use of the thumb-screw under the eye-piece, by means of which the telescope 
can and should be levelled at every observation, the thumb-screw being turned 
up to a bearing, and the plate turned out from under the round-headed screw 
before observations are begun. 

6. Levelling to render the axis of rotation of the whole instrument vertical. 
Bring the telescope and level into a position parallel to the line joining any 

two of the levelling screws, say I. and II., at the base of the instrument. Turn 
them until the bubble is at its middle point. Kotate the instrument until the 
telescope is over II. and III., and adjust III. until the bubble is in the middle. 
Then bring the telescope over III. and I., and the bubble should still be at its 
middle position. If this is not the case the operation should be carefully re- 
peated. Should this fail to correct the difficulty, it shows that the preceding 
adjustment (5) has not been correctly made, or has been disturbed, and must be 
set right. 

7. To focus the cross-wires by means of the eye-piece. 

With the positive eye-piece, such as that in this telescope, the operation of 
focussing the cross-wires consists merely in drawing out or pushing in the eye- 
piece until the wires are perfectly distinct. The best test of the correctness of 



60 ADJUSTMENTS OF THE CATHETOMETER. 

this focus is that when the telescope is carefully focused on some distinct point, 
the wires do not appear to move over this point as the eye is moved from side 
to side of the eye-piece. This adjustment is an important one in all instruments 
using cross-wires. In making it, the eye should be repeatedly taken from 
the telescope for a moment's rest, so that no strain of the eye may cause a 
slightly faulty setting of the wires, which would produce in turn slight inaccu- 
racy, and unnecessary fatigue in the after use of the instrument. 

8. Centering the cross-wires. 

This is necessary in order to bring into coincidence the optical and geometri- 
cal axes of the telescope. Without this adjustment an error would enter when 
the points, whose difference of level is to be measured, lie at different distances 
from the telescope. 

Let the telescope rest in position on its support. Focus it on some minute 
and well-defined point, and direct the intersection of the cross-wires carefully 
upon that point. Rotate the telescope in its supports, — the vertical axis being 
clamped, — and observe carefully whether the intersection revolves about the 
fixed point or remains exactly upon it whatever the angular displacement of the 
tube. The latter will be the case when the wires are in adjustment. If they are 
found not to be so, they must be set right by repeated trials, the adjustment 
being effected by means of the screws near the eye-piece which hold the ring 
bearing the wires. This adjustment requires only an occasional test, and the 
cross-wires should never be disturbed except when out of adjustment. 



MAGNETIC INCLINATION OR DIP. 61 



MAGNETIC INCLINATION OR DIP. 

Read carefully Kohlrausch, " Physical Measurements," pp. 129-132. It is; 
also recommended that the student read Chap. xvii. of Glazebrook and Shaw's 
" Practical Physics." 

Before taking any readings, remove the needle from the instrument and 
reverse its magnetism carefully, as directed at page 132 of Kohlrausch. Then 
study the distribution of magnetization in the needle by placing it on the board 
provided for the purpose, and beneath a sheet of paper, and sprinkling the paper 
with fine iron filings. If any ''consequent points" or irregularities in magneti- 
zation appear, the remagnetization must be repeated in reverse direction. Care 
must be taken to use the same bar or horseshoe magnet each time in remag- 
netizing, and to give the same number of strokes, so that the magnetic moment 
will be as nearly as possible the same in all cases. Be sure to remove all 
particles of iron filings from the needle before replacing it on its bearings. 
Also, always replace both bar magnets in their box, in contact with the 
armatures and with opposite poles together, or if a horseshoe magnet is used 
put on its armature; and remove all magnets to a distance such that they shall 
not affect the needle, being careful not to disturb other electrical or magnetic 
work that may be going on near by. To study the difference in effect pro- 
duced by the magnets when without and when with armature, sprinkle iron 
filings on a sheet of paper laid over the magnet in each case, and observe that, 
with the armature in place, the larger portion of the lines of force pass through 
the armature and disappear from surrounding space. To test whether the 
magnets are sufficiently far removed, read the dipping circle, then turn the box 
of magnets end for end and read again. This reversal should not affect the 
needle reading. Put all the apparatus together on leaving the experiment. 

To set the plane of the mirror in the direction of the magnetic meridian, 
remove the needle to a distance from the instrument. Place the instrument on 
the cardboard base in such a position that the plane of the mirror passes through 
or is parallel to the line marked upon the board, and place a small compass on 
the board with its N. and S. line parallel also to this line, and turn the board 
upon -the table until the line, and thus also the plane, of the mirror are N. and S. 
Level the instrument carefully, read the index on the horizontal circle, remove 
the compass to a suitable distance, and replace the needle on the circle. 

In the instrument used in this experiment, the needle is supported on a steel 
pinion which rolls on agate plates. The friction is so great that without suit- 
able precautions the readings may be several degrees in error. To eliminate the 
effect of friction, care should be taken before each reading to raise the needle off 
its bearings by the screw and lever provided for that purpose, and then to lower 
it again into place, thus insuring its proper placing. After the needle has come 
nearly or quite to rest, the base of the instrument must be lightly tapped (e.g., 
with a pencil) until the needle settles into its final position of rest. The reading 
of both ends (the point just covering its reflection) of the needle should then be 
taken to 0.1°, and the needle again set swinging for a repetition. At least three 
settings should thus be made in each position. The jarring is intended to be 
sufficient to just lift the pivots from their bearings, and thus to nearly eliminate 
the effect of friction. Too forcible tapping will give the needle a throw of suffi- 
cient violence to disturb its position. If the jarring be carefully performed, — 
aud this is the difficult part of the manipulation, — the readings will be quite 
accordant. A series of thirty readings gave an average deviation from the mean 
of less than 0.1°. 



62 USE OF GALVANOSCOPES. 

Three in dependent determinations of the angle of dip i are to be made, and 
the mean value of i computed, using Kohlrausch's formula I. One set -of results 
is also to be computed by II. and III. 

It is important to avoid disturbance of the needle by air currents clue to the 
breath, and to see that the instrument is at a suitable distance from masses of 
iron. Note that the circle is numbered from 0° to 180°, from the bottom to the 
top in both directions. Hence, in order that the angles should correspond to 
the suppositions upon which Kohlrausch's formulae are deduced, a proper correc- 
tion must be applied to the readings. Find this by inspection. . 

Assuming the horizontal component of the earth's magnetism at this place 
to be 0.1717 cgs., what would be the total intensity of the field as deduced from 
the mean result of the foregoing measurement? 



USE OP GALVANOSCOPES. 

In the measurements of many kinds of electrical magnitudes, and particu- 
larly in general practice, there is a large number of methods in which a galvauo- 
• scope of simple and cheap construction may be employed, a galvanometer of 
accurate design and workmanship being unnecessary. Besides this obvious 
advantage, these methods possess, more or less completely, others still more 
important, viz. : 1st, elimination of all constants of the instrument, and thus 
of many instrumental errors; 2d, increased simplicity of procedure and com- 
putation, and greater uniformity of conditions in various steps of the measure- 
ment; 3d, possibility of use of much more sensitive instruments than can often 
be produced when it is necessary (as, e.g., in the tangent galvanometer. See 
p. — .) to adhere to a fixed proportionment of parts. 

Galvanoscopes of great sensitiveness in proportion to their resistance may 
be made, by winding the coils close about a short suspended needle. The appli- 
cation of such will be iudicated in the subsequent notes on Resistance by Substi- 
tution, E.M.F. and B., Resistance of Galvanometers, Wheatstone's Bridge, etc. 

There are two chief functions of galvanoscopes : 1st, to indicate when 
no current is passing in a circuit; 2d, to indicate, with two or more different 
arrangements of the apparatus, the passage of equal currents in a circuit. The 
first case requires simply that the galvanoscope be sufficiently sensitive. 

Neutralizing Magnet. — If the form of coils and needle be not such as to 
give sufficient sensitiveness, this may be very greatly increased by reducing, by 
a neutralizing magnet, the intensity of the magnetic field about the needle. 
This simple process, often overlooked, may render great precision attainable 
with a crude galvanoscope. It is habitually used in the Thomson galvanometer 
and similar scientific instruments. 

Procedure. — A bar or other strong magnet is placed in such a position rela- 
tively to the needle of the galvanoscope that its field at the needle should be 
of the same direction as that of the earth's field at that point, but of opposite 
sign. The resultant field is thus the difference of the two fields, and by varying 
the distance of the magnet, this difference may be made as small as desired. 
The deflection of the needle by a given current is inversely as some function of 
the intensity of this field, so that the sensitiveness increases with the diminu- 
tion of the intensity of the field. The usual. and best position of the galvano- 
scope (see notes on Thomson galvanometer) is with its needle normally in the 
magnetic meridian, and the neutraliziug magnet is then placed with its axis in 
a north and south horizontal line, either in the prolongation of the axis of the 



USE OF GALVANOSCOPES. 63 

needle, or vertically above or below it. When it is necessary to render the field 
very feeble, this may be most conveniently clone by adjusting the position of 
one magnet until the reduction is slightly less than that desired, and then com- 
pleting by another magnet so much more distant that a comparatively large 
movement produces little difference in the field at the needle. 

The reverse of this process is also sometimes useful, i.e., increasing the 
intensity of the field, and thus diminishing the sensitiveness of the galvano- 
scope when too sensitive for a given purpose. 

Discussion. — The needle would be astatic if the field were zero. The 
only special condition in this and the "compensation" process is that the needle 
shall be of so short length that portion of the field through which it moves 
shall be tolerably uniform ; otherwise, there may be points of unstable equilib- 
rium in its motion. 

The second case, i.e., the adjustment of equal currents, may be treated in two 
ways, as follows : — 

a. Equal Deflection Method. — The same current will of course be indi- 
cated by equal deflections on the galvanoscope. The fractional precision of 
this adjustment depends on that with which the reading can be taken by the 
index on the graduated circle, by the spot of light, on the scale, by the 
telescope, or by whatever means is employed, and on the magnitude of the total 
deflection. With the graduated circle (in degrees) and index this fractional 
precision can seldom reach 0.1 per cent, and is often less than 1 per cent. 
With the mirror and spot of light, a long beam of light must be used to get 0.1 
per cent, since the angle of deflection must be small. With the telescope 
0.1 per cent is more readily attainable than with the spot of light. In ordi- 
nary work the index and circle is the most convenient, and often the only 
available means of reading. The neutralizing magnet may be employed in this 
method to extend its range to either greater or smaller currents, but it is not 
generally convenient. The chief difficulty with the equal deflection method in 
general use is, that in order to have the index at a proper point on the scale, the 
limits of strength of current which may be used are quite narrow, which is not 
true of the following method. 

b. Compensation Method. Procedure. — While a current is passing 
through the galvanoscope, a magnet is approached to it in such a position that 
its field at the needle is sensibly equal in intensity, and opposite in direction, to 
that clue to the current. The needle is thus brought back to its initial position, 
and is in a resultaut field sensibly the same as that in which it was before any 
current passed through the coils. If, then, this compensating magnet and the 
galvanoscope remain undisturbed (and also, of course, the direction of the 
origiual magnetic field at the needle), the current passing through the galva- 
noscope will be again the same only when the needle is at this initial position 
(usually, of course, zero), and thus it is possible to determine when the same 
current is again established. It is unnecessary in this method that the galvano- 
scope needle should be originally adjusted precisely to zero, or that the com- 
pensation should bring the needle exactly back to its initial point. The only 
requisite is that the reading of one end of the needle when approximately com- 
pensated should be exactly noted, and the same deflection of the same end of the 
needle exactly reproduced when it is desired to reproduce the current. 

Discussion. — Since the conditions of the galvanoscope are precisely the 
same at both observations, all instrumental errors (except any such irregular 
one as friction) are wholly eliminated. The gain in precision of this method 
results from the possibility of utilizing by it the full sensitiveness of any gal- 
vanoscope with all strengths of current within the limit prescribed by possible 



64 MEASUREMENT OF RESISTANCE BY SUBSTITUTION. 

overheating of the coils. With the equal deflection method, the limiting 
strengths of current which can be used, with a precision of 0.5 per cent, when 
deflections are read to 0.1°, are such as will give deflections between about 20° 
and about 70°, allowing a range of currents of 1:5 at most {i.e., for tangent 
compass). This range can be extended to larger or smaller currents by the 
proper use of the neutralizing magnet, but in the latter case at the expense of 
promptness of motion, and stability of adjustment of the needle. The precision 
of the method is the same with or without the neutralizing magnets. But by 
the use of the same galvanoscope and the compensating magnet (or magnets), 
currents from the above-named lower limit of 20° up to the largest which the 
coils can carry without injury, can be measured with a precision increasing 
with the current. For when the needle is brought to zero by the compensating 
magnets, it will be in such a field that the change of current necessary to pro- 
duce 0.1° deflection will be the same as to deflect the needle from 0.° to 0.1° if 
no initial current were passing. This, in a tangent galvanometer, would cor- 
respond to an error of about 0.5 per cent for a current which would deflect the 
needle to 20°, about 0.2 per cent for 45°, about 0.1 for 60°, and so on. The 
method is thus capable of determining a current which would give a deflection 
above the greatest with which the equal deflection method could deal, and of 
making this determination with greater fractional precision than is possible at 
any deflection on the same galvauoscope, by the equal deflection method. The 
precision increases with the magnitude of the deflection which the given current 
produces on the galvanoscope, and it is, therefore, desirable to have the coils so 
wound as to have the greatest possible sensitiveness with the largest wire which 
can be advantageously used. The strength or number of compensating magnets 
to be used must, of course, increase with the sensitiveness of the galvanoscope 
and strength of current. 

The neutralizing magnets may, of course, be employed to still further extend 
the sensitiveness of the aiTangement. Also it is possible to. make the compen- 
sation by an auxiliary coil with a separate current, instead of by magnets. 



MEASUREMENT OP RESISTANCE BY SUBSTITUTION. 

The simplest method for the measurement of the electrical resistance of con- 
ductors is based on a direct application of Ohm's Law. Suppose a closed circuit 
consisting of a constant battery, a resistance to be measured, and a galvanoscope. 
The latter will show a deflection corresponding to the current passing. If, now, 
the resistance to be measured be removed, this current and the deflection will 
increase, and the insertion of a resistance precisely equal to the one removed 
will be necessary to bring the current again to the same strength, and thus the 
needle to the same reading, as in the first instance. If this latter resistance be 
inserted by means of known resistance coils, then their amount will give the 
measure of the resistance sought. This method is called the measurement of 
resistance by substitution. The process just described is known as substitution 
by " equal deflections," since the current is always brought to the same amount 
by reproducing upon the galvanoscope the original deflection. A second 
process of using the substitution method is by "compensation." 

Read the notes on "Use of Galvanoscopes," page 62, and, in the measurements 
of this experiment, follow the directions there given for both " equal deflec- 
tion" and "compensation" process, using the "neutralizing magnet" if neces- 
sary. 



MEASUREMENT OF RESISTANCE BY SUBSTITUTION. 65 

That the galvanoscope should be sufficiently sensitive to a given percentage 
chauge in the resistance to be measured requires that this resistance should be 
a considerable fraction of the whole resistance, and thus that the galvanoscope 
and battery resistance should not be excessive. The battery resistance should 
be as small as is consistent with the necessary electromotive force, and the 
plate surface should be large, to diminish the effect of polarization. The battery 
employed must be constant. The electromotive force should not exceed that 
which, through the galvanoscope alone, or through the resistance to be meas- 
ured alone, would produce sufficient heating effect to change these resistances 
by an amount corresponding to the precision of measurement desired. The 
ratio of the galvanoscope to the exterior resistance must depend on the sensi- 
tiveness of the galvanoscope in any individual case, being less, of course, for 
the same precision as the sensitiveness is greater. When absolute resistances 
are desired, the temperature must be taken into consideration. (See notes on 
Temperature Correction in Resistance Measurements.) The method is best 
adapted to rapid measurements of moderate resistances, with a precision of one 
per cent to one-tenth per cent, and its main practical advantage is that, with the 
exception of the box of resistance coils, only crude apparatus is needed. A 
Thompson reflecting galvanometer more or less shunted may be used, but any 
short needle sensitive galvanoscope will answer most requirements. 

Kempe's "Handbook of Electrical Testing," Chaps, i. to iii. inclusive, may 
advantageously be read in this connection. Record in note-book diagrams of 
the component and resultant forces producing equilibrium of the deflected 
needle, in the conditions of no compensation, of over, under, and exact compen- 
sation, without the use of a neutralizing magnet ; and also of exact compensa- 
tion with the use of the neutralizing magnet. 

Measure the resistance of the coil C, first by the deflection method, then by 
the compensation method. Then measure each of the other coils A and B by 
the compensation method. In each case make at least three independent 
measurements, and compute the average deviation of each series. Measure also 
Cancl^l in parallel circuit, and compute from the former measurements what 
this should be. (Pickering's "Physical Manipulation," ii. 259.) Also measure 
A, B, and C in parallel circuit, and compare the computed and observed resis- 
tance as before. In usiug the galvanometer, care must be taken to see that the 
needle swings clear. The galvanometer may contain two or more sets of coils, 
one of which is short and of few turns, and is connected between the binding 
posts marked S ; the other is longer, i.e., of more turns, antl is marked L. 
Use in any case the coil which produces a suitable deflection with the given 
resistance in circuit. It will be fouud that with the smaller resistances, the 
long coil may give a greater deflection than can be neutralized by the use of 
even two ordinary bar magnets. In such case the short coil should be used. 
In using the bar magnets, take pains to disturb the work of others as little as 
possible. Leave the magnets in their box, with opposite poles in contact with 
the same armature. If a Daniell battery is used, remove the porous cup aud 
ziuc to the extra jar containing the zinc sulphate solution, at the close of the 
work. 



66 



DIFFERENTIAL GALVANOMETER. 



DIFFERENTIAL GALVANOMETER. 




Fig. 36. 



The differential galvanometer consists of two coils 
of wire of equal resistance, and so placed that when 
the same current is passed through both at the same 
time and in opposite directions, no deviation of the 
magnetic needle suspended within them is produced. 
The wires composing the two coils are usually wound 
side by side on the same spool, or are twisted together, 
and then wound as one wire. The Clark differential 
galvanometer is arranged as shown in Fig. 36. G and 
G 1 are the two coils, so connected that a current enter- 
ing at A and going out at B will pass through G and 
cause the north end of the needle to deflect (e.g., to 
the west). A current entering at D and going out at E 
would then pass through G v and deflect the north end 
to the east. Thus, if the negative (copper, carbon, 
etc.) plate of the battery be connected to C, and the positive plate to Z, upon 
pressing down the key K, the current will flow from C through K to A, thence 
through G to B, through the diagonal wire and plug P to B, thence through G 1 
to E, Z, and back to the battery. If the plug is out, or the key open, the cur- 
rent will not pass. The plug is withdrawn when it is desired to use the coils in 
separate circuits. Set the galvanometer up with the plane of the coils in a 
north and south direction. Try each coil separately, and the two in series, 
noting the direction of the current and of the deflection in each case, and 
whether any deflection, and how much, occurs when the current passes from C 
to Z. Unless this corresponds to a sufficiently small fraction of the whole cur- 
rent passing, and unless the resistance AGB = DG X E very nearly, the resistance 
of x, as determined between A and B, will be different from that found when x 
is between D and E. If this difference is quite small, the mean results will be 
close enough; but if it is not small, the galvanometer requires readjustment. 

The simplest' method of measuring a resistance with the differential galva- 
nometer is to connect between the binding screws A and B, a box, i? 1? of resis- 
tance coils, and between D and E the resistance R 2 to be measured, or vice versa. 
Connect the — plate of the battery with C, and the + with Z. When the circuit 
is closed by pressing the key, the plug being in place, the current passes to A, 
where it divides according to the law of shunts ; part passing through G, part 
through R 15 in the inverse ratio of the resistances. These parts reunite 
at B, flow to Z), there are divided again between G 1 and 2? 2 , reunite at E, 
and flow out at Z. Thus, since the resistance of G is exactly equal to that of 
G v the same fraction of the whole current will flow through each only when 
R 1 = R 2 . Hence to measure R 2 the coils at R 1 are adjusted until no deflection 
occurs upon pressing down the key, when the resistance of the coils is the 
desired resistance. Ordinarily it will be impossible to obtain from the resis- 
tance box such an amount as to just equal R 2 ; in such case the deflection with 
the resistance just too large, and with the one next smaller, may be noted, and 
by interpolation the fraction approximately found. 

Find with the galvanometer the resistance of the wire on the spools I. and 
II. successively. Then connect them side by side to form a "derived," "par- 
allel," or "shunted" circuit, and measure the combined resistance. Compute 
this also by the law for shunts from the previous measurements, (Pickering's 



SLIDE WIRE BEIDGE. 



67 



"Physical Manipulation/' vol. ii. p. 259,) and compare the observed and com- 
puted results. 

The differential galvanometer is convenient for many purposes. A simple 
way of finding approximately the resistance of a battery with it is as follows. 
Connect two (or any even number) of similar cells in opposition. Such a com- 
bination will ordinarily produce a slight current, as the electromotive force of 
one cell will seldom be exactly equal to that of another. This will soon be 
partly overcome if they are allowed to remain on a closed circuit for a few 
"moments. It may also be partially eliminated in other ways, but will be a 
source of less error than that resulting from the polarization produced by the 
current from the main battery connected to C and Z. To measure 2 B, the re- 
sistance of the pair of cells, connect the like plates to D and E, and adjust 
B as nearly as possible, leaving the key down for only a moment at a time. 
It will be noticed usually that upon connecting to D and E, the needle will 
be deflected by the current from the pair of opposed cells. The amount of 
the deflection will change from time to time, and will be either reversed or 
increased according to its direction after the key has been closed for a few 
moments. The change is due to the reverse current sent out by the polarization 
caused by the current from the other battery. 

Measure the resistance of the cells provided for the purpose, and note what- 
ever changes in current and other effects of polarization you may be able to 
observe. This measurement will be found almost impracticable with cells like 
the Leclanche, but illustrates the difficulties inherent in the measurement of any 
liquid resistances, and of the resistance of single liquid batteries. The method 
is quite readily applicable to constant cells. 

Differential galvanometers are often made with the wires coming out to 
three binding posts only. In such an instrument the coils G and G x would be 
the same as in Fig. 36, but with all the connections beyond 
deba removed. The end a would then be brought to one 
binding post, a, Fig. 37, the end d to another, c, and the 
two ends b and e, joined together, would be brought to a 
middle post b. The outside connections for a resistance 
measurement with such a galvanometer might be made, of 
course, to correspond with those of the Clark Differential, 
but they would be generally made as shown in Fig. 37. If 
such a galvanometer is provided, first test the galvanometer 
by connecting / direct to a and h to d. Then measure at x 
each of the coils already measured. Interchange R and x, 
and repeat both. It is convenient to have the key between 
either b or g and the battery. 




SLIDE WIRE BRIDGE, OR B. A. DIVIDED METRE 

BRIDGE. 



Read Kempe's " Handbook of Electrical Testing," pp. 191-204. 

Determine % and n 2 (see § 212), using the coils marked A and B, whose 
resistances are stamped upon them. 

Determine ??i 1 and m 2 (see § 211), using successively for m x and m. 2 the coils 
marked C and D. Use at a the coil marked E (resistance is stamped on it), 
and at x the coil B. 



68 wheatstone's bridge. 

Then using C and D at m 1 and m 2 , measure F, using G as a standard (resis. 
stamped upon it). Also measure A, using B as a standard. (See last formula 
in § 214.) Then remove m l and m 2 , and repeat these measurements with straps 
in at these points (formula in § 212). Note that the bridge readings for which 
Kempe's formulae are arranged are in mm. In each, measurement take at 
least five independent settings on the wire, record each, and find average devia- 
tion from mean. This quantity divided by the length of the shortest, side of the 
wire (including the value of m on that side) will give an approximate measure of 
the precision of measurement, but the errors entering into the measurement of 
n v n 2 , m x and m.,, will, of course, affect any subsequent work involving the use 
of these quantities. 

In the present experiment, various coils of wire made up for the purpose 
are used for the sake of convenience. In general, of course, adjustable resis- 
tance boxes might be employed, and the resistances varied to suit the demands 
of the "Best Conditions ".(§§ 215 et seq.). "When these conditions are carefully 
fulfilled, this instrument becomes capable of a high degree of precision, and, 
for some purposes, is very convenient. The principle of the divided wire is of 
very ready application for a variety of purposes in measurements requiring 
only moderate precision. For many purposes a galvauoscope of low resistance 
and rendered sufficiently sensitive by a neutralizing magnet (see notes on Use 
of Galvanoscopes) may be employed, as in the apparatus provided with this 
experiment. 

WHEATSTONB'S BRIDGE. 

The principle and theory of the bridge may be found in Pickering's " Phys. 

Manip." ii. 260, Ganots' "Physics," § 955, and* must be read before beginning the 

experiment. In the diagram in Fig. 38 let J/, X, 

X, and P, represent four resistances, then the 

points B and C will have the same potential only 

wdien these four resistances fulfil the proportion 

M x 

— = — , and in this case no current will flow 7 

through the galvanometer G when the circuit is 
closed. If J/, X, and P are then known resis- 
tances, the resistance of x will be x = -zz . P. The 

X 

various actual forms of the bridge differ greatly 
from this diagram. The following directions 
apply, except as to details of the apparatus, to the use of any bridge. Record 
the number of the bridge used.. 

The Bridge. — The plugs between A and B and between B and C, Fig. 39, are 
supposed to be removed in all that follows, so that there is no connection between 
these points. The resistances corresponding to M, X, and P, in the resistance 
box LB, are in the lines EB, EFC, and CDHLA respectively. They are in the 
form of double wound coils of wire within the box, each coil having ends 
attached to the brass block on either side of the corresponding plug. The 
resistances in ohms are stamped on the top opposite the plugs. (Details may be 
found in Jenkin's "Electricity and Magnetism," Kempe's "Handbook of Elec- 
trical Testing/' etc.) Thus wdien any plug is removed, the current of electricity 
must pass through the corresponding coil in order to get from the brass block 
on one side of this plug to the next beyond the plug, and there will be thus 
interposed a known resistance. If the plug is firmly in place, its resistance is so 




WHEATSTONE S BRIDGE. 



69 



nearly zero as to be negligible. The coils in the line ALHD are graded like the 
weights in a box of metric weights, and the insertion of resistances by the 
removal of plugs corresponds exactly to and must be proceeded with in exactly 
the same methodical way as the putting of weights into the scale pan of a 
balance. (See "Method of Weighing," p. 7.) The spot of light reflected from 
the galvanometer mirror upon the scale plays precisely the same part as the 
index or pointer of the balance. The lines of resistances il/and N correspond, 



S, JSA 




Fig. 39. 

as will be seen on further inspection, to the arms of the beam of the balance; 
but they differ in this respect, that while the arms of the balance (except the 
steelyard, to which this instrument is even more closely analogous) are fixed in 
relative length and are generally equal, the bridge arms M and N may have 
several ratios. The use of the bridge with various ratios of 31: N is thus analo- 
gous to weighing in a balance in which the arms can be so altered in relative 
length as to make 1 gramme in (e.g.) the right pan counterbalance 10,000, 
1000, 100, 10, 1, 0.1, 0.01, 0.001 or 0.0001 grammes (see numbers on M and N in 
Tig. 37) at will, in the left pan, thus varying the sensitiveness and range of the 
balance enormously at will. 

Galvanometer. — A description of the galvanometer ordinarily used with 
the bridge (Thompson reflecting galvanometer) may be found in Jenkin, 
Kempe, and most text-books on electricity or physics. The proper adjustment 
of the instrument will be made and explained by the instructor at the request of 
the student. The two terminals of the galvanometer G are connected to the 
points B and C respectively of the bridge box. This connection is not direct, 
however, but through a key and a shunt. 

The Shunt is an instrument for diverting more or less of the current passing 
from C to B, so that only a suitable portion shall go through the galvanometer 
itself. This prevents sending an excessive amount through the galvanometer, 
whereby it would often- be injured and the procedure be always rendered 
unnecessarily slow. It would correspond to some attachment to a balance, 
whereby its sensitiveness might be modified at will. The construction of the 
shunt will be understood from reference to the figure and the instrument. The 
current coming from the bridge by the wire C will tend to pass along fh to 
the switch ha; but if this be at the pin a, there will be no further circuit in this 
direction, and the whole current will therefore necessarily go through fC to the 
galvanometer, then back through TcB'gB to the bridge, thus all going through 
the galvanometer, iu which case that instrument will be at its full sensitiveness, 
— a condition in which it is to be used only when the final measurements are to 
be made and a less sensitiveness has been found insufficient. If the switch be 
turned to rest upon the pin e, then a part only of the current will pass through 
G, the remainder through the resistance coils joining e, d, and c, and thence to 



70 wheatstone's bridge. 

g and B. If the switch is on d, only the coil dc will be in circuit; if at c, there 
is only the very small resistance of the switch and connections, so that almost 
the whole of the circuit coming in at C would now be directed through the 
path fhcg to B, and thus almost none go through the galvanometer. The switch 
should be in this position when the apparatus is left at the close of the work. 
The resistance of the coil between d and c is made equal to ■$%■§ of the galvano- 
meter resistance, so that by the law of shunts, when the switch is at d, 0.999 of 
the current passes through fhdcg and only 0.001 through the galvanometer. 
Similarly the resistance between e and c is \ of that of the galvanometer, so that 
with the switch on e, 0.1 of the current only passes through G'. The switch 
should be put upon d at the outset, and always when any change in the resis- 
tances is made which is liable to cause any considerable change in the current 
flowing through G. The switch is changed to e only when sufficient deflection 
of the spot of light is not obtained with it on d, and to a only when the deflec- 
tion is too small with it on e. 

Keys are necessary, as at K and k, in both battery and galvanometer circuits, 
and it is essential that E, the key in the battery circuit, should be closed before 
Jc, and kept closed until after k is again opened. This allows the induced 
currents accompanying the beginning of the currents in the bridge box, 
unknown resistance, and connections to pass away before the galvanometer 
circuit is closed; and the breaking of the galvanometer circuit before the battery 
circuit prevents corresponding induced currents accompanying the cessation of 
the battery current from affecting the galvanometer. To accomplish this result 
easily, the keys are usually combined on a single base. This double key consists 
of an upper leaf of metal connected with one battery wire e, a second similar 
leaf connected with the continuation of this wire m. There is next a third leaf, 
similarly connected to one of the galvanometer wires, and below this a pin con- 
nected with the continuation of this wire. The leaves are all insulated from 
each other. On pressing down the upper leaf, it makes contact with the second, 
and thus closes the battery circuit. Pressing down still further causes the 
third leaf to make contact with the pin beneath it, thus closiug the galvanome- 
ter circuit. On relieving the pressure gradually, the galvanometer circuit is 
first broken, then the battery circuit. 

Procedure. — See that the connections are rightly made throughout, that all 
binding screws are firmly (not violently) turned up, and that all the plugs are 
firmly (not very forcibly) twisted into place. Ask the instructor to see that the 
galvanometer is properly adjusted, and to explain any matters not clear after a 
careful reading of the whole of these notes and references. Connect one end of 
the coil marked B under the upper nut of the post A, Fig. 37, the other end 
under the part at B. Draw the plugs between A and B and between B and C. 
When these are drawn, the connection, formerly made by the plugs, is wholly 
broken. These are called "infinity plugs," as they serve to interpose an infinite 
resistance. Draw the 100-ohm plug in the line EB, and the 100-ohm plug on the 
line FC, thus making M= 100 = N, so that M / N— 100/100= 1, the arrangement 
thus becoming analogous to an equal-arm balance. Thus from the proportion 
M : N= x : P, it appears that for each ohm in x, one ohm must be drawn in P. Place 
the switch of the shunt on d (the one-thousandth shunt). Draw the 1000-ohm 
plug in AL, putting the loose plug in the hole provided for it in the middle of the 
adjoining block. Watch the spot of light on the screen; press the key for an 
instant only (but not violently), letting it up immediately, and note the direc- 
tion in which the spot moves — say to the right. This resistance is probably too 
much. Eeplace firmly the 1000 plug, and try next the 1-ohm coil in the same 
way — suppose the spot to go to the left. Then 1 is too small ; and the spot 



wheatstone's bridge. 71 

goes to the right when the resistance unplugged is too large, to the left when 
too small. Replace the 1-ohm plug, and try the 500, 200, etc., successively, 
exactly as weights in a balance (Notes, p. 7) until it is found, e.g., that 11 ohms 
is too large and 10 is too small. Record the ratio used and the resistance found. 
It is possible, by the deflection readings described later, to get nearer than 
1 ohm with the present equality balance, but in this instance it is not neces- 
sary. 

If it is found during this measurement that the deflections of the spot of 
light become less than about 1 cm. when the one-thousandth shunt is used, the 
switch must be shifted to the tenth shunt, and then, if necessary, to the stop a, 
so that the full current passes through the galvanometer. Care must be taken 
always to return the switch to the thousandth shunt when any considerable 
change is made in the plugs. These directions in regard to the use of the shunt 
apply equally to all the subsequent work with the bridge. It will be seen that 
this variation of the shunt corresponds to a change of sensitiveness of a balance, 
arrangements for which do not exist in any of the usual forms of balance. 

The next step towards obtaining a more precise measurement of the resistance 
is to obtain a balance with a different ratio of the bridge resistances M and N. 
Having returned all the plugs except the infinity plugs firmly to place, take 
out the 1000-ohm plug between E and Paud the 100-ohm plug between E and B. 
The ratio is now M : N= 100 : 1000, or 1 : 10. Hence for a balance P must be 
made = 10 x. The plugs in A, L, H, D must therefore be drawn systematically, 
starting with 110 ohms, using suitable shunts, until it is found, e.g., that 108 
ohms is too much, and 107 ohms is too little. The resistance of x will then be 

between — =10.8 ohms and — = 10.7 ohms. 110 ohms is used at the start 
10 10 

because the previous measurement showed that 11 ohms was too much, and 10 
ohms too little. It would be useless therefore to start with more than 
10 X 11 = HO. And the object in starting with the resistance known to be too 
large is simply as a check to see that the spot goes to the right as before, with 
too large resistance unplugged. If it fails to do so, then either the plugs have 
not all been properly in place (a frequent occurrence with beginners) or the 
coils are not correctly adjusted to the amounts marked upon them (analogous to 
errors in adjustment of set of weights). The latter error can be corrected only 
by a table of corrections which sometimes accompanies the instrument. The 
next place of significant figures may now be obtained in two ways, both of 
which are to be tried. First, by deflections. This method must generally be em- 
ployed to get the last place of significant figures. Bring the spot to the zero of 
the scale, when the key is up, by the bar magnet on the table, or by the neutralizing 
magnet upon the galvanometer if necessary. This and the following manipula- 
tions of the spot will be difficult when jarring of the building or magnetic dis- 
turbance keeps the needle in irregular swinging; in such case the student must 
get the best readings obtainable without spending an excessive amouut of time. 
With P = 107 ohms and the ratio of M : N= 100 : 1000, and suitable shunt, press 
the key down, and hold it down, until the spot comes to its position of rest. 
Record the position of the spot, e.g., 2.5 cliv. to left. Make P= 108, and record 
the position of the spot, e.g., 6.3 cliv. to right. The spot therefore moves 
2.5 + 6.3 = 8.8 cliv. for 1 ohm change in P under these conditions (so that 8.8 
corresponds to the sensitiveness of the balance (Notes, p. 11). If therefore it 

2.5 
were possible to acid — - ohm =0.28 or 0.3 ohm to Pwhen P is 107. ohms, the 

o.o 

spot would be brought up to the zero, i.e., would remain undeflectecl when the 
key was pressed, and the balance would be complete. Thus the result would be 



72 wheatstone's bridge. 

M 100 

x = — . P = — — - x 107.3 = 10.73 ohms. This process is exactly the analogue of 

" Weighing by Swing of Balance," p. 10, and may be carried out in exactly the 
same manner when the galvanometer is upon a sufficiently steady foundation. 
Second, make M= 10 and 2V= 1000, or M= 1 and N= 100. Either ratio is about 
equally good, and that usually preferable which produces the largest deflec- 
tion on the galvanometer in the following work. Make P=1100, and then 

adjust until it is found, e.g., that 1073 is nearly right. Thenar — _x 1073, or 

j to 1000 

— X 1073 = 10.73. By the deflection method, another place of significant figures 

might now be found, using a sufficiently sensitive adjustment of the galvanome- 
ter or increased battery power, but with the apparatus used in this experiment, 
and in any case without introducing correction for the change of resistance of 
the wires with change of temperature, it is useless to carry the result to more 
than four places of significant figures. 

Measure next the resistance of the coil marked A, obtaining as'many places 
of significant figures as possible. Measure to four places the resistance of A 
and B connected in series, and compare with the resistance computed from the 
previous measurements ; also measure them when connected in parallel circuit, 
and compare with the resistance as computed by law of shunts. Measure any 
other resistance provided with the apparatus. 

The most sensitive arrangement of the bridge is with the galvanometer con- 
necting the junction of the two highest resistances of the bridge with the two 
lowest, if G has a higher resistance than the battery, which is the usual case. 
Thus when M and x are greater than N and P, and G greater in resistance than 
the battery, G should be connected as in Eig. 37. The sensitiveness is ordinarily 
greater when M and N are as large as possible consistent with obtaining the 
desired ratio; for this reason it is undesirable to use the ratio of 0.1 : 0.1, 1. : 1., 
1. :0.1, etc. It will, of course, be readily understood that, apart from this 
reason, the precision would be greater with larger coils in 31 and JV, because 
these are more likely to be adjusted correctly to the fourth decimal place, 
and also the slight increase of resistance due to loose plugs in the arms M or 
N would be a less fraction of the total resistance of these .arms. It is also 
evident that the slight increase in the ratio obtainable by making, e.g., N— 1100 
when N= 10 over making N= 1000 when N= 10, would be much overbalanced 

x. « • • *^ 1100 110 1 ■*! 1000 100 rnu 

by the inconvenience of the ratio — — = — — as compared with — — - = ——. The 

infinity plugs at I and J are of service when the box is applied to some other 
uses, and for the present use may be a source of error. 

With a bridge containing the coils shown in Eig. 39, show what would be the 
largest and what the smallest resistance which could be measured to four places 
of significant figures, presupposing sufficient sensitiveness of the galvanometer. 
Note that by changing the battery wire from E to F, the 1000 coil can be thrown 
into M when so needed to obtain ratios suitable for measuring large resistances. 
Give two demonstrations of the formula for the bridge, one by the graphical 
method, one by the use of Kirchoff's Laws. 



TEMPERATURE CORRECTION. 73 



TEMPERATURE CORRECTION IN RESISTANCE 
MEASUREMENTS. 

The temperature of the coils of the resistance box should be measured by a 
thermometer having its bulb inside the resistance box. Precise work cannot be 
done where the temperature is rapidly changing; indeed, absolute measure- 
ments with precision as great as 0.1 per cent require numerous precautions. 
Unless the temperature is nearly constant, it is impossible to ascertain the 
average temperature of the wire in the coils with sufficient closeness. The 
temperature of the wire whose resistance is to be measured must also be taken 
with a precision dependent upon the accuracy sought. Since the resistance of 
most pure metals increases about 0.4 per cent per degree centigrade rise in tem- 
perature, it is obvious that if results to one per cent are desired, no measure- 
ments should be taken without observations of the temperatures at the time. 

Suppose that the resistance IP of a piece of wire at an observed temperature, 
T D , is measured by a bridge (or other resistance coils) whose observed tempera- 
ture is t°; and suppose that these coils are standard at r° (usually marked on 
the box). It is required to find, first, the resistance W l of W at T° in terms of 
the standard ohm, i.e., of these coils at t° ; and, second, the resistance W 2 of 
Wat some chosen temperature, T 2 °, of reference. 

Let p t be the mean coefficient of change of resistance with temperature for 
the measuring coils between 0° and t°, then /3 t — a + bt, in which the values of a 
and b, as obtained by Matthiessen [Jeukiu's "Electricity and Magnetism," p. 
253], are: — 

a b 

Most pure metals +0.003824 +0.00000126 

German silver + 0.0004433 — 0.000000152 

1st. Note that the resistance of the bridge coils at t° would be reduced at 
0° in the proportion of 1 : (1 + & t f). The fixed wire resistance W, measured as W 
apparent units when the bridge was at t°, would therefore have been found as 
W (1 + M) units had the bridge been at 0°. And similarly had the bridge tem- 
perature been t° when the measurements were made, the resistance would have 
been found as 

w = w 1 + B t t standard ohmSt 

1 + P r T 

2d. Let 5 be the mean temperature coefficient for the measured wire (found 
as /3 was) ; then this wire would possess at 0° a resistance in standard ohms, 



At any other temperature, T 2 ° its resistance in standard units would be 



Wi l + ^T 2 _ W 1 + B t t 



1 + 8 T T 1 + /3 t t 1 + 8 T T 

These expressions become simplified when T 2 = 0°, as d T2 T 2 disappears. 
When both wire and coils are of same material, so that 0=8, and when 
0, W 2 becomes 

vtr _ W 



If t and t are but 2° or 3° apart, 
1 + A* _ 



1 + 7 T (P — T ) approx. 



74 SPECIFIC RESISTANCE. 

where y T =a + 2 /3r = the "true coefficient" of change of resistance of the wire 
with change of temperature. Most resistance coils are made of Germau silver. 

If the work is of precision not much exceeding 1 per cent, the change of 
resistance may be allowed for by adding or subtracting 0.4 per cent per degree 
for most pure metals, and 0.04 per ceut for German silver. 

Prepared tables of resistances of copper aud other substances at various 
temperatures are given in most electrical handbooks (Wigan's "Electrician's 
Pocket Book," p. 165), and are convenient for ordinary purposes. 



SPECIFIC RESISTANCE. 

The experiment on Wheatstone's Bridge must have been performed before 
this is begun. Read also the following references before beginning work : 
Kempe, "Handbook of Electrical Measurements" (3d ed.), pp. 166 to 172 
inclusive, stating in your record the conclusions arrived at in §§ 196 and 197; 
pp. 31 to 43, but do not disturb the adjustment of the galvanometer, as prac- 
tice in setting this up is given in connection with the work in measurement 
of electrostatic capacity; pp. 10 to 15; Jenkin, "Electricity and Magnetism 
(7th ed.), chap. xvi. §§ 14 to 17; Wigan, "Electrician's Pocket Book " (2d year's 
ed.), pp. 162 to 165. 

In using resistance coils, great care should always be exercised to avoid 
sending excessive currents through them, and to keep the circuit open at all 
times when it is not essential that it should be closed. Otherwise the heating 
caused by the current may seriously impair the insulation of the coils, and also 
change their resistance. In the present experiment, and always when a pre- 
cision exceeding one per cent is desired, the temperatures of the coils and of 
the wires measured must be taken and proper corrections made as described in 
the notes on "Temperature Corrections," p. 73. 

Measure the quantities necessary for finding the specific resistance at 0° C. 
of all the samples of wire provided, in terms of both the centimetre cube and 
the metre-gramme. Compute these, and also for the copper wire, its specific 
conductivity (Jenkin, p. 252), and its percentage conductivity referred to pure 
annealed copper as given by Matthiessen. 

In the measurement of the specific resistance, to obtain the length of the 
wire, stretch it between two nails placed in the floor at as great a distance 
apart as the wire will permit. Mark a point near each end, and measure the 
length between these points. Insert the wire in the bridge in such a manner 
as to measure the resistance between the same points. To obtain the tempera- 
ture of the wire as nearly as possible during the resistance measurement, coil the 
wire upou the paraffined board or other arrangement provided. Place the ther- 
mometer so that its bulb shall be against the wires. Read the temperature, 
which is to be corrected for instrumental errors of the thermometer, while 
the resistance measurements are being made. Read also immediately after the 
measurements the temperature of the bridge coils, inserting the thermometer 
into the hole provided for this purpose in the vulcanite top. 

The weight of the wire is obtained by weighing the whole wire and making 
proportionate allowance for the end pieces, whose leugth must therefore be 
measured. The diameter of the wire must be carefully measured at twenty or 
more points along its length with the micrometer gauge, correcting for zero 
error. The mean may be used, although as the resistance varies inversely as the 
square of the diameter, the use of the mean square would be more exact, and 



COEFFICIENT OF TORSION". 75 

it should be employed if the differences in diameter are considerable. Avoid 
bending the wire sharply or winding it into a small coil. Aim at precision of 
0.1 per cent in the final result. 

Compute from the results obtained for copper the error in each of the com- 
ponent measurements made, which would produce an error of 0.1 per cent in 
the final result, and give this calculation in the record. 



COEFFICIENT OF TORSION. 

Any suspending fibre when twisted must exert a torsional resistance. In an 
instrument of a definite precision, either this resistance must be proved negli- 
gible, or must be measured aud corrected for. One method is here given of the 
determination of the torsion correction in a suspended galvanometer needle. 

Set up the instrument with as little initial torsion as possible. Read both 
ends of the index in their zero position. By means of a magnet or otherwise, 
with as little strain upon the suspension as possible, and without disturbing the 
position of the instrument as a whole, give the needle one complete rotation. 
Read both ends again. The difference from zero reading will give the angle a x 
through which the torsion deflects the needle from its initial position. Rotate 
the needle back to its initial position ; read again ; twist one rotation in the oppo- 
site direction, and read. This will give a r Make a similar series of readings 
with two rotations in each direction, and then with three. Compute from these 
a mean value a of the deflection for one rotation of the needle. The silk fibre 
is somewhat viscous, and the effect of this in a gradual yielding of the fibre is 
perceptible when the angle a is large or when the fibre has been many times 
twisted. This renders the correction uncertain in some cases. Note any effects 
of this sort that are observable when 10 rotations are used. 

The angle through which the fibre must be twisted to produce a is 860° — a . 
The couple which held this torsional force in equilibrium was that produced by 
the action of the horizontal component H, of the intensity of the earth's mag- 
netic field on the poles of the suspended needle, aud this is 

Hml sin a, 

where m — strength of pole of needle, and I = distance apart of the poles of the 
needle, and this quantity is, therefore, equal to the moment of torsion in the fibre. 
By laws of torsion, the moment of torsion due to a twist of the fibre through 1° 

would therefore be Hml — — , and for any twist of <f>° it would be $ times this 

ouO — CI 

quantity. This fraction, — — > is called the " Coefficient or Constant of Tor- 

obU — CL 

sion," aud will be here denoted by 0. 

The application of 8 in the use of galvanometers will be illustrated by the 
case of a standard tangent galvanometer. (See notes on "Tangent Galvano- 
meter," and "Law and Factor of Galvanometer.") For that instrument, the 
equation of equilibrium between the force due to a current passing through the 
coils, the earth's magnetism, and the torsion of the fibre, calling C the current 
in cgs. absolute units, / 

C %™ ml cos <b = Hml ( sin 4> + <i>. s]na ) 
r Y { r r 360 -a/ 

= Hml (sin <p + <p . 0). 



76 LAW AND FACTOR OF GALVANOMETER. 



Hence c= jr_. Bing + g.g ; 



2irn\ 



tan <p + 



2tt)1 COS <p 2tt)1 \ cos <p 

2irn \ siu <j 

or, in general, for any tangent galvanometer 

V=Kttm<b(l + -&^L 

\ sin <p 

For small deflections <p, expressed in circular or radian measure, = sin $, 

approx., so that the parenthesis becomes simplified to (1 + 6 f ), where d' = , 

2tt — a 

the value of a being also in circular measure. This simplification often serves 
also where 6 is so small as to be nearly vanishing. Note that to reduce 6, the sus- 
pension may be either lengthened, or made finer, or the magnetic moment of the 
needle may be increased. To determine whether or not 6 need be used in a par- 
ticular instrument, consider that if it is desired to introduce into C from neglect 



of torsion, no error exceeding, e.g., 1 in 1000, then as the term 1 + ~t — enters 

V smf/ 

as a direct factor in the computation of C, it must not exceed in amount 
(1 + 0.001). 

Compute the percentage error which would be introduced into C from neg- 
lect of the torsion correction for the fibre and needle used in this experiment, if 
they were in a tangent galvanometer, and the deflection was $ = 60°. 



LAW AND FACTOR OF GALVANOMETER. 

The student must be familiar with the demonstrations regarding the tangent 
galvanometer, the unit of current, etc. (Notes on " Coefficient of Torsion," p. 
75, Jenkin, viii. § 2; Gray, " Abs. Meas. in Elec." Chaps, iii. and iv., etc.) 

Form of Standard Galvanometer. — A simple and convenient form of 
standard tangent galvanometer for measurements to 0.2 per cent or thereabouts, 
of currents from 0.01 up to 10 or 20 amperes, may be made by the suspension of 
a magnet or bundle of magnets of about 1 cm. in length, at the centre of a ver- 
tical circular coil of one or more turns of wire, the radius of the coil being from 
two to ten decimeters, and its cross-section small in proportion to its radius. If 
the coil consists of several turns, a sufficient length of wire should be measured 
before winding, then carefully and regularly wound into a proper channel, and 
measurement taken of any portion of this length removed during construction. 
The resistance should be measured before and after winding, and a proper use 
of paraffin should insure freedom from defective insulation. If the length is not 
thus known, it must be determined by sufficiently careful measurement of the 
diameter or circumference of the coils during or after winding, and a correct 
count of the number of turns in each layer of the coil. Several details of con- 
struction are given in Gray, Chap. iv. 

Galvanometer Constant and Factor. — When the dimensions of the coils 
and magnetic needle, and the torsion of the suspending fibre, are of proper magni- 
tudes, the current in amperes passing through the coils of such a galvanometer 
may be computed by the formula, 



LAW AND FACTOR OF GALVANOMETER. 77 

where H — horizontal component of the intensity of the earth's magnetism, 
I = total length of wire in the coils, 
r = mean radius of the coils, 
n = total number of turns of wire in the coils, 
<p — observed mean angle of deflection of needle. 

In this expression it is known that —„, or — — , gives the intensity of 

v 1 r 

field produced at the centre of the coil by a unit current passing through the 

coil. This intensity is of course a constant for any individual galvanometer of 

whatever form, and in standard galvanometers with circular coils the quantity 

_, or— , may conveniently be denoted by the' term "Galvanometer Con- 
T i r 

stant." Following Maxwell's nomenclature, this will be represented by G. Thus, 
the above formula becomes 

C = i°^. tan*. 

G r 

It will be seen that the sensitiveness of the galvanometer is proportional to 
G. The galvanometer constant can be computed from the coil dimensions in a 
few simple forms of coil only, of which that already described is most frequently 
convenient. In the last preceding expression it will be seen that the whole frac- 
tion — — — , i.e., } is constant for each instrument so long as H remains the 

G 2 irll 

same, a condition ordinarily nearly fulfilled when the galvanometer remains fixed 
in position, and no disturbance by masses of iron, magnets, or electric currents 
takes place in its vicinity. Thus, in the ordinary use of a galvanometer, this 
quautity is a constant factor, by which it is necessary to multiply some function 
of the observed angular deflection {e.g., the tangent) in order to determine what 
current (in amperes) produced the given deflection. Thus in 

C = ^-^ tan <p = K. tan <p, 

K may be called the "factor " of the galvanometer in question. If G can be 
computed, and H is known, of course K can also be computed, and this is the 
case with the standard tangent galvanometer. For this instrument, the follow- 
ing expression serves to give results to 1 part in 500, provided that the various 
quantities are determined with sufficient accuracy, that the dimensions of the 
cross-section of the coil (supposed rectangular) are small (say not more than \ 
to j 1 ^) compared with the radius r, and that the construction and adjustments 
are correct : 



2irn \ 2 f 2 3 r 2 J\ 4 f 2 A r' 2 j\ sin 

The parentheses give respectively : 

I. Correction for dimensions of sections of coils : 

2 b = breadth of section (parallel to axis of coil), 
2 d = depth of section (parallel to plane of coil). 
II. Correction for length of needle : 

2 I = length of needle between magnetic poles. 
III. Correction for torsion of fibre : 

= coefficient of torsion, see p. 75. 



78 LAW AND FACTOR OF GALVANOMETER. 

Any or all of these corrections may become negligible by proper propor- 
tioning of parts, and the above expressions are used for computing the 
proportions. 

Find the coefficient of torsion, and compute and record for the standard gal- 
vanometer used (note its number), the numerical value of each correction, for 
deflections of 30°, 45°, and 60°, and apply (by interpolation) in every ease 
throughout this experiment those whose omission would introduce 'an error of 
1 in 1000. The proportions between b, d, r, 7, etc., may be computed for any 

given value of r, by equating the terms -— — - — ; —-.— + 12...E sin 2 *; and 

2 r 2 3 r 2 4 r 2 4 r 2 

^r— » successively to the limiting fractional error, e.g., 0.001, and solving for 

-' -j -; 0, etc. The first will give also value of - to make error zero, by equating 
to zero, and solving for -• (Maxwell, "El. and Mag." ii., 317; Kohlrausch, 

" Phys. Meas.," 154 to 157.) 

The index should be secured rigidly to the needle, and parallel to its axis. 
An index at right angles to the needle is necessary in a cosine galvanometer on 
account of clip, but is undesirable in a tangent galvanometer, on account of the 
error introduced where the index is not exactly at right angles. With a circle 
of 4 to 6 inches' radius divided into degrees, a fine index may be read to 0.1° with 
an error of less than that amount, so that the mean angle readings described below 
will have an average deviatiou of perhaps 0.05°, and are therefore precise to 
that extent, disregarding undetermined errors of graduation. For very small 
deflections the mirror and telescope are required. The inertia of the needle 
should be small, and the directive force of its magnet as great as possible, the 
needle being best made up of several strips of steel strongly magnetized. - 

In the tangent galvanometer the angle of greatest precision of current meas- 
urement is 45°, where, as may be seen from a table of natural tangents, the error 
produced in the tangent by 0.1° error in reading is 0.0035, and as tan 45° = 1, 
this is 0.35 per cent. So that the precision of the mean reading at 45° would be 
about 0.17 per cent by the previous assumption. The preeision is the same 
at equal distances on each side of 45°. Errors of 0.05° at any points between 30° 
and 60° make less than 0.2 per cent, and between 20° and 70° less than 0.35 per 
cent in the tangent, so that as far as the influence of errors of reading is con- 
cerned, deflections ranging anywhere from 30° to 60° give about the full precision 
of which such an instrument is capable. 

To Adjust the Galvanometer. — 1. Raise needle until it swings clear. Do 
not twist fibre in so doing. 

2. Level by foot screws, until the plane of the coils is vertical (within 0.2°), 
judging by a level. Plane of circle should be originally equally nearly at right 
angles to plane of coils. Turn about vertical axis until needle is nearly in plane 
of coils, thus putting this plane approximately into magnetic meridian. All 
disturbing masses of iron must be removed from vicinity. Then repeat levelling 
if necessary. 

3. Eccentricity. — Centre needle approximately in graduated circle, if auy 
adjustment of suspension is provided, by making both ends read nearly the same 
when uncleflected, also when deflected to about 70° by a current. Do not use too 
strong a current. In reading deflections, place the eye so that the index exactly 
covers its own image. 

Middle of needle should be in centre of middle plane of coils. Also adjust 
circle so that the line of the zero points is as nearly as possible in the middle 
plane of the coils. 



LAW AND FACTOR OF GALVANOMETER. 79 

4. Meridian Adjustment. — Turn coils on vertical axis until index is as nearly 
as possible parallel to line of zero points, thus completing 2. Level again if 
necessary. 

To Read the Galvanometer. — 5. Always read both ends of needle to elimi- 
nate eccentricity. Call reading a x and a 2 . 

6. Always reverse current, and read a 3 and c^. 

7. Then <]> = g i + a 2 + a s + a 4 e ii m i ua tes eccentricity, error in adjustment of 

zero line, error of setting coils in meridian, error from initial torsion; the three 
last, however, only approximately, but sufficiently when the error is small. 
Throughout this experiment it is supposed that the standard galvanometer can 
be read to 0.1°, and that it is desired to eliminate all other errors separately 
exceeding 1 in 1000. 

Liaw of Galvanometer and Determination of Factor. — In a galvanometer 
whose coils are not designed with proper proportions between parts, or in such 
a shape that the corrections above given can be computed, it may be important 
to determine what law connects $ and C, and if this function is nearly that of 
tan $, to measure the deviation from that function, so that the factor and cor- 
rections to obtain C from <p may become known. This may be done in many 
ways, of which the two most general are here given. The first is an application 
of Ohm's law; the second requires the use of a standard galvanometer (supposed 
in what follows to be also a tangent galvanometer). 

By Ohm's Law. — Note always the numbers or marks of all instruments used, 
and draw sketch of circuit as actually arranged. Connect in direct circuit a 
constant battery (which should have large plate surface to reduce polarization 
changes), a known variable resistance, and the galvanometer to be studied (not 
the standard in this case, although that instrument might of course be tested in 
this way). Give the resistance successively such values R v R 2 , etc., as will pro- 
duce deflections of about 15°, 23°, 35 D , 45°, 60°, 75°, or as nearly this frequency 
and range as possible. Then take the series with the same resistances in reverse 
order. Average the corresponding mean deflections, and use the resulting angles 
as $ 1? <p 2 , etc., in the subsequent work. If there are differences in the corre- 
sponding mean angles of the two series, these will be in large measure due to 
changes in the battery during the experiment. If they exceed 0.5°, the battery 
must be further inspected, and the series repeated. The battery should be put 
on short circuit for a few minutes before beginning the observations. 

1. If the E.M.F., E, and resistance B, of the battery were known, and also 
the resistance of the galvanometer 67, and connecting wires r, then the current 

causing <p x would be C-, = — - -~ , and C might thus be computed for each 

JO + J*l + «T -f- T 

value of (p. A plot with values of C as ordinates, and <p as abscissas, would then 
serve for interpolation in subsequent use of the galvanometer. This process is, 
however, not suitable in most cases, since E and B are subject to considerable 
uncertainty. Also, if the galvanometer followed the law of tangents (tested as 
follows), the galvanometer factor If could be determined from these values of 
C and <p in a manner to be described later in the case where C is measured by a 
staudard galvanometer. This method may be omitted from the present work. 

2. Since the usual test to be made is as to whether the deflections follow some 
supposed law in regard to the currents, e.g., C being proportional to <j>, to tan <p, 
to sin <p, or in general to any function F((f>), the problem is simplified. Let 
the supposed function be tan <£, so that the supposed expression is 

C = Kt&xicp, 



80 LAW AND FACTOR OF GALVANOMETER. 

where Kis the factor of the galvanometer. Also, in the above observations, 

E 



G = 



combining these gives 



B + B + G + r ' 
E 



B+B+ G+r 



Kta,n<p, 



where, as E and K are constant, {B + R + G + r) is proportional to and 

tan <p 
therefore to cot <p. If, therefore, a plot were made with values of {B + B + G + r) 
as abscissas, and of cot <p as ordinates, the result would be a straight line only 
when the galvanometer followed the law of tangents (within errors of observa- 
tion). But as B is often unknown, and it may be inconvenient to measure G and 
r, the plot is to be made with the observed values of B as abscissas, and of cot <j> 
as ordinates. This line, it will be seen, should be identical with the one just 
described, except that the axis of cot <p (ordinates) is shifted by an amount 
(B+(? + r) to the right. If, therefore, the straight line (if such be obtained) be 
prolonged to intercept the axis of resistances, the intercept will give (B+G + r). 
If G is known (Wbeatstone's Bridge), (B + r) may be found by subtraction, and 
hence B, since r is usually negligible. In general, the plot would be made with 

as ordinates. If E were known, K would be given by multiplying the tan- 

± (<?>; 

gent of the angle between the line obtained and the axis of resistances by E. If 
a Daniell cell is used, assume E — 1.08, and compute K. Give demonstration of 
this method of finding K. "When two galvanometers are used, as in the subse- 
quent readings, they must be so far apart that a test shows that the strongest 
current passing through either does not affect the other. 

Obtain from the plot the point of greatest divergence from the law of tan- 
gents between about 30° and 60°, and state the percentage error at this point in 
terras of the whole tangent. A more precise estimate may be made of the devia- 
tion, by computing for each observed deflection <p the product R tan <p, where R 
is the total resistance in circuit. This product should be constant, and its per- 
centage variation is readily found. Compute the maximum percentage variation 
between about 30° and 60°. 

By Comparison with Standard. — "When a standard galvanometer is avail- 
able, a better test is the following, which should next be made. Connect in series 
the galvanometer to be studied, the standard galvanometer, the battery, and the 
resistance box. The same current will now flow through both galvanometers. 
If the deflection of one is more than twice as great as that of the other, the 
greater should be reduced by shunting that galvanometer with such a resist- 
ance in the form of a wire or a resistance box (see Jenkin, xvi. § 3, or Kempe, 
" Handbook of Electrical Testing," Chap. iv.), as will make the deflections 
about equal. If the shunt is employed, the following directions as to com- 
putation must be modified, as will be seen by consulting the above refer- 
ences. Vary the resistance until the readings of the galvanometer to be tested 
are about 10°. 20°, 30°, 40°, 50°, 60°, 70°, 80°, successively, and read each time 
both galvanometers. Reversals of currents in both instruments and readings of 
both ends of needle are always essential. Compute the factor of the staudard 
galvanometer, as directed at page 77, assuming H= 0.1717, and compute the 
current C for each deflection of the standard galvanometer. Plot a curve 
with mean deflections of the unknown galvanometer as abscissas, and cur- 
rents as ordinates. This curve would serve for interpolation to give the current 
corresponding to any deflection of the galvanometer, if no simple law connected 
the deflection and the current. To test whether the galvanometer follows the 



LAW AND FACTOR OF GALVANOMETER. 81 

Law of tangents, plot a series of points with tangents of its mean deflections as 
abscissas, and currents as ordinates. If the law of tangents is followed, these 
points will lie along a straight line (otherwise the line will be curved) two of 
whose properties are of interest. Call K the factor of the unknown galvano- 
meter, supposing it to follow the law of tangents, then O=j5Ttan0, where 
is the deflection produced by the current C. Now, when = 45°, tan 0=1, 
and thus C = Kx 1. Hence, the ordinate of this line at the abscissa 1 (i.e., cor- 

C 
responding to tan 45°) is the factor of the galvanometer. Also, since K= - — -, 

iT is also given by the tangent of the angle which the line makes with the hori- 
zontal axis. If the work is properly done, these two results will of course agree 
within the limit of precision attainable on the curve, which should be plotted on 
a large scale. 

When the second galvanometer is already known to follow the law of tangents 
or other function, a single comparison with the standard may of course suffice 
to determine its factor. 

In this method, as in that based on Ohm's law, a more precise discussion than 
the graphical may be had by computation. If the tested galvanometer follows 
the law of the tangents, and be its deflection corresponding to $ on the stand- 
ard, then 

tan 



tan<£ 



a constant. 



Compute, therefore, this ratio for each deflection of the series, and from the 
ratios find the maximum percentage deviation between 30° and 60°, as before. 

If the galvanometer tested is one (suppose a tangent) of a possible precision 
of 1 iu 500, i.e., about equal to the best standard tangent and to the possibil- 
ities of reasonable certainty in regard to H, the foregoing graphical methods are 
not sufficiently precise. Iu such a case, the comparative observations having 
been made as just directed, let the deflections of the standard be represented 
by <p, of the unknown by 0*. Then 

„ 10 Hr x ^ x 

C = - • tan <p = iTtan 

Find the mean value of 




2-rrn tan 

from two or more pairs of observations (those in which and <f> are symmetri- 
cally disposed about the 45° point are preferable). Then compute the corrections 

10 Hr 

tan <p x — K tan 9 X = d v 



2jtW 

10 Hr 



tan 9 — iTtan 9 = cL, etc., etc., 

for all the observations, and plot values of as abscissas and of d as ordinates, 
on a large scale. Then, in subsequent use of the tested galvanometer, 

C=Ktan9+d, 

d being found by interpolation on the curve, at the abscissa 0. This method is 
capable of much refinement and of several modifications to adapt it to special 
cases, e.g., to commercial iustruments of various forms. 

From the definition at page 77, it will be perceived that if K be determined 
by any of the preceding methods for any galvanometer whatever, and H be 
known for the time and place, then the " constant " G for that galvanometer can 
be computed once for all. G is a fixed quantity for that particular instrument 



82 E.M.F. AND R. OF BATTERY. 

so long as its coils, needles, etc., retain the same relative positions, sizes, etc. ; 
and thus the factor would be known subsequently for the same or any other 
place if the value of H for that time and place were known. 

By Electrolysis. — The determination of the factor K may also be made by 
electrolysis, as will be described in the notes on the electrolytic determination 
ofil. 

Sine Galvanometer. — The standard tangent galvanometer, if provided with a 
vertical axis of rotation, should now be used as a sine galvanometer (see Kempe, 
- Handbook of Elec. Testing," chap. iii. ; Wigan, "Electrician's Pocket-Book," 
p. 71), by putting in circuit with it the battery and resistance box. Adjust R 
until the deflection is $ = 30° to 40°, and read as a tangent galvanometer. Then 
turn the coils about the vertical axis, and read as a sine galvanometer, first on 
one side, and then on the other side, of the zero, with the north end of the needle, 
then similarly with the south end. Let the mean of these four readings be 6. 
Then tan cp should equal sin 9. It should be observed that the sine galvanometer 
as thus constructed affords no means of correcting, for want of exact adjustment 
of needle to plane of coils, except by reversal of current and entire re-setting of 
the instrument. It will be found, in general, much less convenient in use than 
the tangent galvanometer, though possessing some advantages. 



E.M.F. AND R. OF BATTERY. 

The following method is applicable to the so-called " constant " cells only, as 
■would be at once shoAvn by trial with a Leclanche or other inconstant cell. 
Even with the more nearly constant cells there is sufficient change in the electro- 
motive force E, and resistance B, of the cell during use with varying external 
circuits to cause the measurements to be somewhat unsatisfactory. This is not 
to be ascribed to a fault of the method, but to difficulties inherent in any battery 
in which a liquid electrolyte is used, the polarization effects in liquids being 
often very troublesome, and at best only partially avoidable. The smaller the 
currents used in the measurement, and the less their duration, the less the diffi- 
culty from polarization. 

Connect in direct circuit a Daniell battery, a taugent galvanometer of known 
resistance G, and known factor K, and a resistance box, R, by which resistance 
from 0.1 ohm or 1 ohm up to 100 ohms may be inserted. Make the external re- 
sistance R + G as low as possible, without obtaining a deflection greater than 
75° on the galvanometer, and read the deflection a v calling B x the resistance 
inserted. Then the current is 

a = Kttm «, = — 

1 * B + B + G x 

the resistance of such amount as to 
Then 

Co =K tan a„ = 



B + R 2 +G 

Increase R again, and obtain smaller deflections, a 3 and a 4 , in succession. 
These four deflections should range from about 75° or less, down to about 15°, 
or as nearly that as the conditions will allow. It is sometimes desirable to give 
to the ratio of the successive currents the value 2 : 1, or some special value to cor- 
respond to conditions of some case of practice. 

Assuming the E and B to remain constant in any successive pairs of meas- 
urement, they may be computed as follows. Compute both in this way from 



E.M.F. AND E. OF BATTERY. 83 

each successive pair of deflections a^ and a 2 , and a 2 and a,, etc. Follow care- 
fully the directions given in the notes in regard to the adjustment and use of 
the galvanometer, pages 78 and 79. 

The factor it is first to be computed from the dimensions of the coils of the 
galvanometer (unless otherwise specified ou the instrument). From this the 
current C v C 2 , etc., corresponding to the deflections a v a 2 , etc., are of course 
found from the expression C x = K tan a v etc. Then, by Ohm's law, denoting 
by r v r 2 , etc., the total resistance Z? x + G + «?, i.e.. the sum of the inserted resist- 
ance, that of the galvanometer, and that of the connecting wires (usually neg- 
ligible), E-Cr 
E=C 1 B+C l r v .: B= Lj 



Also, E=C 2 B + C 2 r 2 . ;.E = ^-{E- Cyj + C.,r 2 . .-. E= C " C ^ r,f ~ r ^ . 

From this value of E two values of B may be computed by substitution in the 
two equations E= CB + CV, from which the numerical values of E were de- 
duced. The mean of these two values may be used as a value to correspond to 
the value of E. But a better way of finding B is this. Equating the first two 
expressions for E gives 

C^B+rJ^C^B + r,). 

C\-C 2 ' 

from which values of B may be deduced corresponding to the above values of E. 
Inspection will show that this contains in both numerator and denominator the 
common factor K, since O t = A" tan a v etc. Striking out this factor leaves 

B. 



tan a x — tan a 2 

which is independent of K. Thus, the resistance of a battery may be deter- 
mined by means of a tangent galvanometer (or any galvanometer whose function 
is known) without a knowledge of K. 

The values of tan a are in all cases to be corrected for torsion of the sus- 
pending fibres (see notes on " Coefficient of Torsion," p. 75). 



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